Estimating Detection Limits in Ultratrace Analysis. Part II: Detecting and Counting Atoms and Molecules

feature articles

Estimating Detection Limits in Ultratrace Analysis. Part II: Detecting and Counting Atoms and Molecules*

CHRISTOPHER L. STEVENSONt and JAMES D. WINEFORDNER:~ Department of Chemistry, University of Florida, GainesviUe, Florida 32611-2046

Although the limit of detection (LOD) is a widely used means of indi- cating the detection power of an analytical technique, the application of conventional detection limit theory is not straightforward with respect to laser spectroscopic methods which are capable of detecting single atoms or molecules in the laser beam. In this paper, theoretical considerations in the detection and evaluation of these methods are addressed on the basis of a simple model of a typical laser spectroscopic experiment; in addition, the theoretical requirements for the precise counting of atoms, and factors which influence the signal variance in the model, are considered. Simple Monte Carlo computer simulations are used to verify and demonstrate the application of the theory of single atom detection (SAD) to typical experimental situations.

Index Headings: Detection limits.

INTRODUCTION

The limit of detection (LOD) is an important figure of merit (FOM) which is designed to give an indication of the detection power of an analytical technique by specifying the lower limit of analyte concentration or amount that can be distinguished from the background noise. The importance of the LOD value is especially evident when one is evaluating and comparing ultratrace techniques, which can detect a vanishingly small amount of analyte in a sample. Improvements in methodology are usually aimed at achieving a higher detection power; thus, the comparison and improvement of these ultratrace methods will usually involve LOD values, and it is important to fully understand and carefully evaluate the nature and limitations of this figure of merit.

This three-part series of papers is intended to explore various aspects of detection limits, especially for laser spectroscopic methods capable of detecting a very low amount of analyte. In Part I of this series, 1 the variability of the estimated LOD of a general analytical procedure was addressed and applied to a model based on a very sensitive laser spectroscopic method, laser-induced fluorescence of analyte atoms in an electrothermal atomizer (ETA-LIF). In the final two installments, the concern is

Received 14 October 1991. * This research supported by DOE-DEOFG05-88ER13881 with partial

support by Eastman Kodak (Rochester, NY). t Current address: Advanced Monitoring Development Group, Health

and Safety Division, Oak Ridge National Laboratories, Oak Ridge, TN 37831-6101.

~: Author to whom correspondence should be sent.

directed to methods in which the sensitivity is high enough to detect a very low number of analyte species (atoms, molecules, etc.) within the laser beam during the measurement, even down to the ultimate limit of the detection of a single analyte species in the laser. The application of the LOD concept to these methods is not always obvious, but the evaluation of the methods needs con- sistent and appropriate definitions specifically designed to address the difficulties which result from the ability to detect small numbers of atoms.

Single-Atom Detection (SAD) Methods. The use of lasers as sources in spectroscopic techniques often results in a unique blend of extremely high sensitivity and se- lectivity. As a result, many ultratrace methods are based on the detection of processes which result from the interaction of analyte species with a laser beam. Indeed, there are now laser spectroscopic methods which claim the ability to detect single analyte atoms or molecules; the first reports of the detection of individual atoms/ molecules appeared well over a decade agoY ~ Laser-based methods in which the sensitivity is high enough (and the noise is low enough) to ensure that individual species in the laser beam can be detected shall be called single- atom detection (SAD) methods. This terminology follows the convention established by Alkemade, 5 where detected analyte species are collectively referred to as "atoms," even though the analyte species may actually be atoms, molecules, ions, or radicals. SAD methods as described in this paper all deal with the detection of events produced as a result of the interaction of the atom with the laser beam. Techniques capable of SAD can be broadly categorized as using either destructive or nondestructive methods of detection. In the former class, the atom is consumed during the detection process, producing at most one single "count" per atom; in the second category, each atom can produce multiple detectable events during its interaction with the laser.

In this paper, attention will be focused on two laser- based methods which have had the most success at detecting single atoms: resonance ionization spectroscopy (RIS) and laser-induced fluorescence (LIF). The first method involves the laser-assisted ionization of the analyte atoms and subsequent detection of the ion or elec- tron in a buffer or a vacuum; this processes is usually destructive. Since the production and detection steps in such processes can both approach 100% efficiencies, it

Volume 46, Number 3, 1992 0003-702S/92/4603-040752.00/0 APPLIED SPECTROSCOPY 407 © 1992 Society for Applied Spectroscopy

is not surprising that there have been a number of reports of the detection of a very low number of atoms with the use of RIS2 ,4,~4 Two books detailing the various methods and approaches to RIS have been recently published. 15,'6

Laser-induced fluorescence methods detect the spon- taneous emission of laser-excited analyte atoms; there have been a number of claims of SAD using LIF. 2,17-34 Methods which have achieved SAD using LIF are usually based on a "cyclic" interaction of atom and laser, and each atom produces a relatively large number of photons during its interaction with the laser; thus "cyclic" LIF is an example of SAD using nondestructive detection.

Although there have been some reports of SAD in which individual atoms in the laser have had a high signal-to-noise (S/N) ratio, 2,17 most of the past SAD methods have had a comparatively low S/N ratio for each atom; in these cases there is often a question of whether or not single atoms can actually be detected above the background (if any background is present). Evaluation of these methods often involves the following questions:

1. Can individual atoms (SAD) be detected reliably? 2. Under what conditions is SAD possible? 3. If it is possible to detect single atoms, is it also possible

to count the numbers of atoms passing through the laser beam during the measurement?

Indication of the detection power of these (possible) SAD methods, through the application of detection limit theory, is not a straightforward task. In comparison with practical aspects, theoretical considerations for SAD methods have received very little attention in the liter- ature. There have been a handful of papers which have evaluated laser spectroscopic techniques with respect to the potential to detect atoms; 23,3~-39 these include some theoretical discussion of SAD methods. Recently, several papers have attempted to deal with the problem of ver- ifying that single molecules were being detected by LIF as they flowed through the laser beam. 3°-33 However, Alkemade's approach remains the only in-depth, system- atic, general theoretical treatment of SAD to date; this work is presented in two classic papers, ~,4° and has recently been reviewed and extended. 41

The purpose of this paper is to provide a basis for answering the questions posed earlier which may arise for methods which produce a signal for single atoms which might be detectable above the background noise. The theory presented is based on the application of LOD theory to a general SAD model; the intent of this treatment is to generalize Alkemade's work and to present a theory of SAD in a form useful in the development and comparison of methods which may achieve the goal of detecting and counting atoms. Computer simulations help to demonstrate and verify the concepts presented here; further application of the SAD theory to the particular problem of monitoring atoms by LIF as the atoms cross a continuous-wave (cw) laser beam will be presented in Part III of this series.

THEORY OF SINGLE ATOM/MOLECULE DETECTION

Before we proceed further, the exact definition of an SAD method, as used in this work, should be clearly

stated. A method can be considered capable of single atom detection only if each and every atom which interacts with the laser can be detected above the background noise. This definition will be re-stated in a more rigorous form later; nevertheless, the general concept of an SAD method can be appreciated. Implicit in the above statements is the assumption that we are concerned only with the analyte atoms which actually interact with the laser, and not necessarily every analyte atom present in the sample. Equally important is to notice the difference between a method in which some (but not all!) individual atoms can be detected above the background. In an SAD method, it is not enough to simply have a certain likelihood of detecting single atoms, but every atom which is probed by the laser must be detected for a method to achieve SAD.

SAD Model: Typical SAD Experiment. The general form of a laser-based SAD method can be illustrated with the aid of Fig. 1. There are a number of terms which will be used to describe the SAD model, some of which will now be defined in reference to this figure; the symbols for these terms are collected in Table I for future reference. In the figure, analyte atoms flow past a region of interaction with a laser beam; atoms which interact with the laser produce a number of detectable events. The number of events detected during a measurement time Tm are counted. During this time, a certain volume of sample containing analyte, Vo, flows past the laser beam; for the analyte atoms within this volume, there is a probability of entering a region in which it is possible to interact with the laser beam and produce detectable events. This region is the probe volume, lip, and is defined in Fig. 1 by the region of intersection of the flowing stream with the laser beam which can be viewed by the detector. Depending on the duty factor of the laser, a certain number Np of atoms which enter Vp interact with the laser beam during Tin; a given atom interacts with the laser beam for time tl. The atom's interaction time, ti, is a function of various factors such as the magnitude of Tin, the type of laser used (pulsed, continuous, or mod- ulated), and the atom's residence time, tr, within Vp. The value of tr is usually a variable for different atoms, with mean rr, depending on such factors as velocity, diffusion, and size and shape of Vp. The atom's corresponding value of ti may also be a variable, with mean ri depending on the relative magnitude of Tm and Tr, and the conditions of the experiment.

The experiment shown in the figure demonstrates one possible form of an SAD experiment, the case of analyte atoms flowing through Vp (e.g., an atomic beam experiment). There are of course various other possibilities for SAD experiments, such as: (1) the detection of gaseous atoms in a heated cell filled with buffer gas; (2) the detection of analyte molecules in a sample contained in a cuvette; and (3) the detection of ions present in an ion trap. In the first two cases, the analyte species might freely diffuse in and out of V~ during Tin; in the last case, the analyte species may all be held in Vp for the duration of the measurement. In some SAD experiments, Np may be fixed during Tin, while in others N~ may change during the measurement; likewise, tr and ti might be constant for all atoms during Tin, while in other cases these values may vary from atom to atom.

408 Volume 46, Number 3, 1992

l a s e ~

analy te f l o w I

ilillll . . . . . . . ::?.:.:. d e t e c t e d

/ ~ . . . . . . . . . . :.. events

volume

Fro. 1. Example of a typical SAD experiment .

Signal Production. Dur ing the m e a s u r e m e n t t ime T~ there m a y be a cer ta in (variable) n u m b e r of b a c k g r o u n d counts , Ib, wi th a m e a n given by

Ub = ~bbT~ (1)

where

~b b = m e a n flux of b a c k g r o u n d coun t s (count /s) , and #b = m e a n b a c k g r o u n d coun t s dur ing T~ (counts) .

T h e to ta l signal, I t , r eco rded dur ing Tm is due to the con t r i bu t i on of noise a nd ana ly te signal. For nondes t ruc - t ive de tec t ion , this can be given as

I , = I b + ~ i s (2) Np

where

i, = n u m b e r of de t ec t ed events due to each indiv idual a t o m which flows t h r o u g h Vp.

Th i s var iable has a m e a n given by

;s = ¢, t l (3)

where

O~ = m e a n flux of signal f r o m a given a t o m (coun t s -~ atom-~).

T h e m e a n to ta l signal for a des t ruc t ive t e chn ique is given by

L = #b + N , ( 1 - e-*~t0 (4)

where the t e r m in pa ren thes i s gives the p robab i l i t y t h a t a given a t o m will be ionized dur ing its in te rac t ion wi th the laser.

I n our SAD model , the ana ly te signal for each a t o m en te r ing V,, is, will be a s s u m e d to be due to a Po i sson process; in addi t ion , it will be a s s u m e d t h a t the l imit ing noise is due to b a c k g r o u n d sho t noise (hence, Ib follows a Po i s son d is t r ibut ion) . T h u s , the p r o d u c t i o n of a signal due to a given a t o m which in te rac t s wi th the laser dur ing T~ is a to ta l ly r a n d o m process. For nonde s t ruc t i ve m e t h - ods, such as cyclic L I F , this means t h a t the n u m b e r of coun t s per a tom, is, is a Po i s son var iable in t ime, and t h a t the signal coun t s are r a n d o m l y d i s t r i bu ted t h r o u g h -

TABLE I. List of some symbols used in the SAD model and their definitions.

Symbols Definitions

(;jo

i,

N,

Nm

N,

RSDc RSDm Tm ti

tr y~ Vo X, x~

~d

(~b

#b

O't 2

Ti

Tr

Number of background counts observed in a measurement (counts).

Number of signal counts from a particular atom (with given ~b, and ti values) during a measurement (counts/ atom).

Number of signal counts from any atom entering laser during a measurement (counts/atom).

Minimum sensitivity requirement for counting atoms at or better than a pre-defined precision level (counts/atom).

Total number of counts (background and signal) observed during a measurement (counts).

Number of atoms which interact with the laser during a measurement.

Number of atoms measured from I, for a measurement (i.e., best estimate of Np).

Number of detected ions (due to analyte only) in an RIS experiment during a measurement.

Precision of counting Np atoms during a measurement. Signal precision due to Np atoms during a measurement. Duration of a single measurement (s). Interaction time of a given atom with the laser beam in

the probe volume (s). Residence time of a given atom in the probe volume (s). The volume probed by the laser beam which can give rise

to counts at the detector (m3). The volume of analyte which flows through V, during a

measurement (m3). Signal detection limit (counts). Guaranteed detection limit in signal domain (counts). Probability of false positive (type I error) during a mea-

surement. Probability of false negative (type II error) during a mea-

surement. The detection efficiency, the probability that an atom

which enters Vp will give rise to a signal detectable above the noise.

The mean flux of background counts (counts/s). The mean flux of signal counts due to a given atom in Vp

(counts/s). The mean signal flux of any atom which enters Vp

(counts/s). The mean number of background counts during a mea-

surement (counts). The signal variance of I, during a measurement (counts~). The mean interaction time of an atom with the laser in Vp

(s). The mean residence t ime of an a tom in Vp (s).

ou t tl. Since Ib is also a s s u m e d to follow a Po i s son dis- t r i bu t i on wi th m e a n of Ub, t h e n when a single a t o m passes t h r o u g h Vp dur ing Tin, It has a Po i s son p robab i l i ty dis- t r i bu t i on wi th a m e a n of ~b + esti. Fo r a des t ruc t ive de t ec t ion m e t h o d , such as RIS , the de tec t ion process is also a Po i s son process, a l t h o u g h in a more subt le way. T o u n d e r s t a n d this, cons ider a large n u m b e r of a t o m s Np all in te rac t ing wi th the laser b e a m at one t ime, re- su l t ing in a m e a n flux ~b~ of de t ec t ed ions. For a Po i s son (des t ruc t ive) de tec t ion process, the de tec t ion t imes of these ions are a s s u m e d to be u n i f o r m r a n d o m variables , and the t ime in terva ls be tween de tec t ion follow an exponen t i a l d i s t r ibu t ion . 42

I f the a s s u m p t i o n of a Po i s son de tec t ion process is valid, t h e n the p robab i l i t y t h a t a t least one single even t will be reg is te red by t ime t~ can be f o u n d by in tegra t ing the exponen t i a l d i s t r ibu t ion f r o m t = 0 to t = ti:

P ( O <- t <<- t i ) = 1 - e - ~ , " (5)

APPLIED SPECTROSCOPY 409

where, as before, q~, represents the flux of counts due to one analyte atom. The source (and conditions for validity) for the term in parenthesis in Eq. 4 is now apparent. Equation 5 is valid for any Poisson detection process, whether the detection is destructive or nondestructive. For RIS, the above equation gives the probability that any atom which interacts with the laser for time t~ will produce an ion that is detected; for cyclic LIF, the equation gives the probability that at least one of the emitted photons will produce a photoelectron above the discrim- inator level. Note that the background noise has not yet been considered; thus, even though the single atom might produce one count (or more), it has yet to be seen whether this can be distinguished from the counts due to the background--i.e., whether or not the atom can be detected.

The meaning of ~b s depends on the SAD technique used. For a nondestructive cyclic LIF, ¢s can be taken as the mean flux of photons from a given atom. For RIS, a "flux" of ions is obviously not possible with destructive detection when N o = 1 atom; in this case, ¢~ is the reciprocal of the mean time for detection of an ion produced from a single atom. Theoretical expressions for q~, for a number of cases for LIF and RIS can be found in the litera- ture.39,4L43,44

Probability Distribution of Signal. As explained above, the signal is comprised of two parts: the background, assumed here to follow a Poisson distribution at the shot noise limit, and the signal, which is due to a Poisson detection process. We will now briefly discuss the probability distributions of It in the case of destructive and nondestructive detection.

For destructive detection, the mean analyte signal due to N o atoms irradiated during T~ was given in Eq. 4. The probability distribution of N~, the observed counts due to N o analyte atoms (neglecting the background), will follow a binomial distribution:

(Np)[1--exp(--~f l i )]N' P (N i )= Ni

X [exp ( - - dp f l i ) ]N . -N ' (6)

where the binomial probabil i ty of "success" is taken from Eq. 5. Equation 6 gives the probability distribution of signal ions with No constant; however, even ignoring the additional contribution of the distribution of background ions, the distribution of signal counts from measurement to measurement will most likely not be a binomial distribution since N o would vary from measurement to measurement. For example, if N o follows a Poisson distribution with mean No, then N~ will also follow a Poisson distribution with mean/~/o(1 - exp[-~b~t~]). Obviously, for RIS, when only one atom crosses through V o during the measurement, then the signal due to the analyte is either zero (not detected) or one count (detected).

For nondestructive detection, the total signal I, due to background and N o atoms is given by Eq. 2. Since both Ib and is are assumed to be Poisson variables, it appears that It should also follow a Poisson distribution:

(it)1'exp(-lt) P(I,) = (7)

It! with mean and variance given by

i t = ~2 = CbTm + No¢, t , . (8) The distributions in Eqs. 6 and 7 give some indication

of the probability distribution of the signals when N o atoms cross the laser during a measurement in an SAD experiment. However, these distributions are strictly only valid when both q~s and ti are equivalent for every atom which interacts with the laser. Although this may be true, depending on the experimental conditions, it can easily be the case that both ~s and ti are variables, with means ~8 and ri, respectively. The values of ¢8 and ti for a particular atom may depend on the path of that atom through V o. For example, in an LIF experiment, q~s depends upon the optical collection efficiency, and this may not be constant over the entire probe volume; another source of variation in ~bs with path might occur if the laser intensity is not constant throughout Vp. For the situation in Fig. 1, the interaction time, t~, of an atom will not be constant if diffusion effects play a significant role during T~, or if the shape of V o is such that an atom traveling down the center of V o will have a larger interaction time than one which skirts the edge.

The effects of variable ¢s or ti will be discussed for the case of nondestructive detection in order to provide some insight into the possible consequences of this variation. Although the number of detected photons, i8, from any particular atom (with certain values of q~s and ti due to its particular path through V o) which interacts with the laser, will follow a Poisson distribution with a mean of ¢st~, the overall probability distribution of counts due to single atoms entering V o will be denoted by the variable Is, with a mean given by

i 8 = +,r+ (9)

and Is will not follow a Poisson distribution if Cs or t~ values are variable. The mean of the total signal can thus be written for the general LIF case:

L = CbTm + NoCsr,. (10)

The effect of having either ~b, or t~ variable is to increase the variance, at 2, of It. The value of at 2 (for a fixed value of N o) can be partitioned between the variance due to the Poisson detection process in both the background and signal counts (shot noise) and the "extra" variance due to any variability in ¢~ and t~:

at 2 = i t + (No)2a2(¢,t,) (11)

where the second term in the equation is due to the "extra" variance. Of course, when Cs and t~ can both be assumed to be reasonably constant, then this "extra" variance term approaches zero and/~ will be (approximately) Poisson.

The above discussion of "extra" variance was limited to nondestructive detection; however, similar considerations would be expected to hold for the signal variance with destructive detection as well: variable ¢8 or ti would tend to inflate the signal variance. The effect of the variable t~ on at 2 will be demonstrated later for LIF using computer simulations.

Signal Detection Limit for the SAD Model. With the SAD model and definition as given, the application of conventional detection limit theory is as follows: For a given measurement time T~, the distribution of Ib is


Poisson with mean and variance #b. The signal detection limit X~ is thus set according to a pre-defined tolerance (denoted by a, the probability for type I error) for false positives:

P(Ib > X~) = ~. (12)

An acceptable value of ~ must be chosen before the experiment; the value of Xd is set so that the observed probability of one or more false counts during Tm (due to background noise) is at or below this level. Note that, since Ib is a discrete variable, the value of ~ will not be uniformly decreased by increasing the value of X~; instead, a will also assume discrete values with different Xd, If Ib is a Poisson variable, then estimating the mean ttb during Tm will allow Xd to be set through the use of tables of the Poisson probability distribution. 45

If it is found that, for given values of Tm and a,

PUb > 0) < ~ (13)

then there is essentially no background noise during Tin; this is called the in tr ins ic noise l imit , since the only noise on the observed signal is due to variance of the signal itself. At the intrinsic limit, a single count (or more) indicates the presence of analyte; i.e., the value of X~ is one count.

Detection Efficiency of a Near-SAD Method. The detection efficiency, ~, of a given atom which interacts with the laser can be generally defined as the probability that the atom will result in a signal de tec tab le above the background noise:

~d =- P( I t >-- Xd) (14)

where X~ is chosen for given values of Cb and T~, according to a pre-defined value of a. At the intrinsic limit, the detection efficiency is limited by the noise inherent on the signal itself. Since we have assumed a Poisson detection process, we can see from Eq. 5, according to the above definition, for tLb = 0 counts, the probability that an atom entering V, will be detected is given by:

cd = 1 -- exp(-q~,ti). (15)

This equation is equivalent to the one used by Alke- made, 4° who was only concerned with the intrinsic- limited case. Essentially, the above equation gives the probability that any atom (for the use of either destructive or nondestructive detection) of given ¢, and t~ will produce a detectable event (i.e., at least one count). How- ever, when there is a significant background present during T~, then this equation does not give the detection efficiency as defined by Eq. 14, since single counts during T~ would be indistinguishable from the background noise level; hence, atoms producing only one count will not be detectable.

Requirements for SAD Methods. The general definition of an SAD method, given earlier, is that the method detects each and every atom which interacts with the laser with near certainty. This requirement is now given in a more succinct form: an S A D m e t h o d is a m e t h o d in wh ich

~d >- 1 - /3 (16)

where/3 is the probability of type II error (false negative).

0.30]

°1°1 IdY '°"

t ,lllli ~.. o.12

o.o6 LIF

cool . , ...... . , , , , , I l l@h, , , . ...... 15 20 25 30 35 40 45 50 55 60 65 70 75

N u m b e r o f c o u n t s

Fro. 2. Probabi l i ty d is t r ibu t ions of a single m e a s u r e m e n t for the signal coun t s for RIS and LIF expe r imen t s wi th Np = 25 a toms and ed = 0.90. T h e RIS signal follows a b inomial d is t r ibut ion, and the LIF d is t r ibut ion is a Poisson. I t is a s s u m e d t h a t ¢, and ti are cons t an t (and equivalent) for all 25 atoms.

The highest allowable value of/3 must be decided prior to the evaluation of the (possible) SAD method.

The application of the above general requirement for SAD is different in the cases of RIS and cyclic LIF. The difference between the two as SAD methods can be best understood by studying Fig. 2 for fixed and equivalent values of Np and ~. The intrinsic-limited case is shown in the figure, and c~ can thus be calculated by Eq. 15. It is assumed that both cs and ti are constant for all Np atoms in both cases (such a situation is reasonable for an atomic beam experiment with a pulsed dye laser). The figure shows that equivalent detection efficiencies would nevertheless result in far different distributions of signal counts, depending on the type of detection process used.

With a destructive method such as RIS, a single atom can only give rise to one count at the most. The only way in which the requirement for SAD will be met as set forth in Eq. 14 is if X~ = 1 count (i.e., the intrinsic-limited case). Thus, for true SAD using RIS (or any other destructive technique) the following two conditions must hold:

P U b > O) < a

exp(-¢ , t i ) </3.

It is important to realize that, when the background counts during the measurement are significant, there is no way to differentiate between counts caused by the blank and those due to a single atom passing through the laser (in fact, in such a situation, the detection efficiency of single atoms, as defined, is essentially zero). Note that the second condition, which follows from Eqs. 15 and 16, assumes constant ¢, and ti. The effect of variable values of these parameters on the overall detection efficiency must be taken into account if necessary.

For nondestructive detection, it is possible to achieve SAD in the presence of noise as long as the following condition holds true: with N, = 1 atom,

P(I t > X , ) > 1 - /3

where/3 is pre-defined for given values of Tm and q~b. The mean of the distribution of It for which P U t < Xd) =


T A B L E II. The two limits for an S A D experiment, with respective type I and type II errors. The limits are given for the signal domain, with all the signals given as counts. For each value of t ~ the limits were chosen so tha t a,/~ < 0.0014 (first row) or a,/~ -< 0.05 (second row). The values in brackets were found by approximating the background with a normal probability distribution.

~ X~ (100%)a Xg (100%)fl

0.00 1 <0.1 6.6 0.14 1 <0.1 3.3 4.98

0.05 2 0.12 8.9 0.14 1 4.88 3 4.98

0.25 4 0.01 12.7 0.13 2 2.65 4.8 4.87

1.00 6 0.06 16 0.14 4 1.90 7.8 4.85

5.00 14 0.07 28 0.13 10 3.18 16 4.33

10.00 22 0.07 39 0.12 16 4.87 24 3.44

100.00 132 [130] 0.13 [169] [0.14] 117 [117] 5.22 [136] [5.00]

is called the guaranteed detection limit, X~, in the signal domain. If both the background, Ib, and the signal,/8, are described by Poisson probability distributions, then it is easy enough to assign values to Xd and Xg for any value of gb by using tables of Poisson values. 45 Table II shows these values for a number of cases, and different values of a and ~. The procedure in determining these signal detection limits is as follows: From the value of ~b, Xd is chosen from the probability tables so that P(Ib > Xa) ~ a. From this value of Xa, a Poisson distribution is found such that P(I~ > Xa) ~ 1 - ~. The mean of this distribution is Xg.

If it is assumed that both ¢, and t~ are constant, then Xg can be found from Poisson tables as described; the requirement for SAD by cyclic LIF can be written

[, > X~ - ~bT~. (17)

Figure 3 shows a situation with ub = 1 count where SAD is possible by LIF detection (a ~ ~ ~ 0.0014). When ~b = 1 count during Tin, Xa = 6 counts and Xg = 16 counts; thus, by Eq. 17, SAD is possible with a sensitivity of 15 counts/atom. Note from Table II that, even in the intrinsic-limited case, a value of 6.6 counts/atom is needed for SAD by LIF at the 99.86 % confidence level, while 3 counts/atom would be needed for SAD at the 95 % level.

It should be emphasized that it is possible for a given technique to detect individual atoms and still not be a true SAD method; for example, in an LIF experiment with ~b = 0, if the sensitivity is 1 count/atom, then individual atoms would still be detected quite often (ca = 0.632 if/8 is a Poisson variable). This is an example of a near-SAD method, where detection of an atom in the laser beam during Tm is possible (and perhaps likely) but not certain.

Precision of Counting Atoms.$ Thus far we have discussed the likelihood of detecting single atoms. However, the question remains whether it is possible to precisely count the number of atoms which pass through the laser beam in a single measurement time, T~, if Np > I atom. In a single measurement with fixed N., the signal dis-

$ I t is a s s u m e d t h a t ~b~ and t~ are cons t an t t h r o u g h o u t th i s section.

0 .50 - @ 0 t-

O.40-

0 0 0 0 .30 -

0 .20 -

0 .10 -

a . 0 . 0 0

m X b Xd

ct,= 15 counts /a tom

II. .., mmmmm . . . .

0 5 10 15 2 0 2 5 3Q

Number of counts

/ N = 0 ~ N = 1

Fro. 3. G u a r a n t e e d de tec t ion of a single a t om in the presence of noise by LIF. T h e background d is t r ibu t ion descr ibes the s i tua t ion when no a tom passes t h rough Vp dur ing the m e a s u r e m e n t ; when N . = 1 a to m the d i s t r ibu t ion of I t h a s a m e a n of Xg and is a lmos t cer ta in to give rise to a de tec table signal.

tributions in the RIS and LIF experiments are given by the binomial and Poisson distributions, and by substi- tution of the appropriate values for a from these distributions, the following equations are obtained for the signal distributions due to a fixed value of Np for the intrinsic-limited case:

RIS: RSD _- ~ / ( _ _ ~ , ) 1 - ~ (18a)

1 LIF: RSD - (18b)

V~ ,~s t i

From Eq. 18a, it can be calculated that for a signal precision of 10% or better (at all values of Np) with RIS it is necessary that the condition e~ >- 0.99 be met. Since the RSD improves as Np increases for RIS, an RIS method with this detection efficiency or better is capable of counting atoms for any value of Np.

The situation for LIF is different, however, since it is possible that many events can be detected from a single atom. Substituting Np = 1 atom in Eq. 18b results in a requirement of 100 photoelectrons/atom for RSD = 0.10 with N~ = 1. Thus, it would seem that, for precise counting of atoms with LIF, a sensitivity of at least 100 photoelectrons/atom during ti is required at the intrinsic limit. This is a far more stringent requirement than the 6.6 photoelectrons/atom which are necessary for SAD (at the 99.86% level).

There is a problem, however, when the RSD is calculated according to the above formula that stems from the difference between the precision of signal measurement, now termed RSDm for LIF, and the precision of counting atoms, RSDc, when a nondestructive SAD method is used. This problem does not arise with RIS since, when SAD is possible, it is essentially true that every atom gives rise to a single count; thus, the signal exactly follows the number of atoms.

Consider the situation for LIF with 100 photoelectrons/atom, shown in Fig. 4. This figure illustrates the difference between the signal precision and the counting


0 . 4 0

0 . 3 2

0 . 2 4

I~1 I

0 0 . 1 6

@ T , = I O 0 c o u n t s / a t o m

2

4 5 6 0 . 0 8 7

i 0 . 0 0

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0

N u m b e r o f c o u n t s ( p h o t o e l e c t r o n s )

FIG. 4. The distribution of signal counts for values of Np from 1 to 10 atoms. The RSD of the individual distributions is given by Eq. 18b. The discrete distributions are represented here as continuous distributions, for clarity.

precision. Although by Eq. 18b this situation corresponds to RSDm = 0.1 for Np = 1 atom, it is apparent that there is very little possibility of incorrectly counting atoms when Np = 1, since there is almost no overlap between the distributions. Obviously, the value for RSD~ is not a reflection of the counting precision of the LIF method.

This discussion will now be concerned with nondestructive (i.e., cyclic LIF) detection only, since the requirements for precise counting by RIS are essentially the same as the requirements for SAD. For LIF, with an observed signal It for a single measurement of duration T~, the value of Np is estimated by N~, according to the following equation:

where the RND function rounds the expression in the parentheses to the nearest whole number, and the integer Nm is the number of measured atoms (i.e., the estimate for Np). We can define the counting precision, RSD~, as

~(N~) RSD~ - (20)

where a(N~) is the standard deviation in the number of measured atoms with fixed Np.

The difference between RSD~ and RSD~ can be seen in Fig. 5, which shows the situation with #b = 0 counts, {, = 20 counts/atom, and Np = 5 atoms. The probability distribution of the signal is shown, with the top axis providing the corresponding values of N~ which would be calculated at a given signal value using Eq. 19. The dashed lines in the figure show the portions of the probability distribution which would result in values of 4, 5, or 6 atoms for Nm. Notice that Arm is an integer; thus, a signal of 75 photoelectrons, for example, would result in a value of N~ = 4 atoms (and not 3.75 atoms!) as the best estimate for the true value of Np.

In order to investigate the counting precision of an SAD method, the theoretical value of RSD, for various fixed Np must be determined at various values of IZb. Calculation of RSD~ would also allow us to calculate the minimum sensitivity, (/s)~, necessary to achieve a desired

M e a s u r e d

0 1 2 3 0 . 5 0 , , ,

0 . 4 0

~ 0 . 3 0

0 0 . 2 0

0 . 1 0

0 . 0 0 ' ' ' ' ' ' ' ' ' ' '

N u m b e r o f A t o m s

4 5 co 7

0 20 40 60 80 1 O0 120 140 160

Signal (Counts)

Fia. 5. Demonstration of the difference between the measurement precision of the signal, RSDm, and the counting precision, RSDc. The portion of the distribution of I, between the dashed lines would result in the corresponding values of Nm given on the top axis. See text for further discussion.

counting precision at all levels of Np. For example, it might be desirable to know what sensitivity is needed to count atoms with at least 10% precision. The theoretical value of RSDc is not as easily calculated as for RSDm in Eq. 18b. To do so, an expression for a(Nm) in Eq. 20 must be found for given conditions of Np, sensitivity and noise. From the definition of the variance of a discrete variable, 4s it is known that

o o

cr2(Nm) = ~_~ ( N , , - Np)2P(N,,) (21) Nm =0

where P(Nm) is the probability distribution of Nm. This probability distribution can be written as

Xu

P(N,,) = ~_~ PUt) (22) It ~ X t

where the summation limits, X~ and X, , are found as follows:

For N= = 0:

Xt=O

Xu = Xa - 1.

For Nm = 1:

Xl = Xd

X, = INT[(Nm - 0.5)¢sti + tzb].

For Nm > 1:

X / = INT[(Nm - 0.5)¢stl + lzb] + 1

X, = INT[(Nm + 0.5)q~st~ + tzb].

In all cases the INT function represents the integral part of the expression in the parentheses.

The above expressions are tedious to solve manually; a computer program can be written to evaluate these expressions (for various values of Np, lzb, and Cs), and so determine ({~)c, the minimum sensitivity required for RSDc _< 0.10. The results, and comparison of the theo-


laser

I

r c a p i l l a r y

w a l l

I I V p I I

I rp i

x p

r l

a n a l y t c f l o w

v \ l

~p

r ,

c a p i l l a r y

w a l l /

z

L_,

(a) (b) FIG. 6. Basis of the Cylindrical Probe model used in computer simulations. The analyte's position is uniformly distributed along the x-y plane, v, is constant and fixed, and vx = vy = 0.

retical value with the observed value of RSDc from computer simulations, will be presented later.

In any practical situation, the value of RSDc would be almost impossible to determine since N , would not be fixed but would be a variable that would change for different measurements (according to a Poisson distribution for the model presented in this chapter). Neverthe- less, this discussion, and results which will be presented later, illustrates that, for a nondestructive technique such as LIF, the requirement of SAD is not sufficient to count atoms during Tm to an arbitrary precision (unlike the case with a destructive SAD method). There is an additional increase in sensitivity required before this is possible by LIF.

EXPERIMENTAL

General. Monte Carlo computer simulations of simple analytical experiments were used as a means for demonstration and verification of the theoretical work set forth in the last section. The simulations were also help- ful in the formulation of the theory which has been presented. The intention of experiments based on these computer simulations is not to perform an exhaustive study of all aspects of the models discussed, but merely to prove their validity and provide some insight into their usefulness. This section will outline the general form of some of the programs which were used in these simulations; more details will be given when appropriate.

All simulations were carried out on IBM-compatible microcomputers which were equipped with either a 25- MHz 80386 CPU and an 80387 math co-processor chip, or a 20-MHz 80286 CPU with an 80287 co-processor. All programs were wri t ten and compiled in Microsoft QuickBASIC (Version 4.5, Microsoft Corp., Redmond, WA). Algorithms to generate random numbers according to normal distribution and Poisson distributions were written according to guidelines presented by Knuth 47 and are based on QuickBASIC's pseudo-random number generator.

In Alkemade's original work on SAD theory, 5,4° the possibility of detecting single atoms in the presence of noise was ignored, and an SAD method was considered possible only in the absence of noise (i.e., at the intrinsic noise limit). In this work, a more general model broadens the definition of a true SAD method to include the presence of noise; however, SAD is only possible under these conditions by a nondestructive detection technique. The computer simulations which will be presented emphasize this fact by modeling an SAD method based on cyclic LIF of atoms flowing through a cw laser beam. All simulations can be classified into two types, according to the method used to calculate the residence time, ti, of an atom within Vp; these models will be called the Simple and the Cylindrical Probe models. In all cases, it was assumed that ~, was constant throughout Vp.

Simulations Based on the Simple Model. In simulations based on the Simple model, it was assumed that the value of tr, and hence ti, is constant for all atoms which enter Vp. In this situation, the variable I, is a Poisson variable, and any "extra" variance is zero. Such a situation can occur with pulsed laser experiments or in situations where the shape of Vp is such that the analyte atoms have (almost) constant tr values for any path through V p - - e.g., if a cylindrical lens was used to focus the laser in a line across the analyte stream, or the analyte flow was constricted so that it all flowed through the center of an unfocused or expanded laser beam. The use of spatial masking to shape Vp might also decrease the variation in C to a very low value.

Simulations Based on the Cylindrical Probe Model. The purpose of this model is not necessarily to provide a more realistic simulation of typical experimental conditions but to demonstrate the effect of a non-zero "extra" variance such as shown in Eq. 11. In this model, the extra variance is due to a variable interaction time with the laser beam. The model is based on an experiment such as was shown in Fig. 1, in which the probe volume as defined by the laser beam and the collection optics (with a slit as a limiting aperture) is a cylinder.

In this model, it is assumed that tr and ti are variables that depend on the path of an individual atom through Vp. The situation used in the simulations, along with labeled axes, is shown in Fig. 6. Effects due to diffusion, flow profiles, and velocity distribution are neglected; all atoms enter Vp with the same velocity in the z-direction, vz. Both v, and vy components of the velocity are assumed to be zero. The Cartesian coordinate system shown in Fig. 6 has its origin in the center of Vp; if the position (x ,y) of the atom within Va is such that x <- x , and y < rp, then this atom interacts with the laser beam. For an atom which enters Vp, the value of ti is dependent only on the position along the y-axis (since v, = Vy = 0).

The interaction time of an atom which enters Vp can be easily calculated for the Cylindrical Vp model according to

2 t, = - - ~v/(r p 2 - y2). (23)

Vz

Note that ti depends only on the uniformly random position of the atom along the y-axis. It is assumed that Vp is small enough for y to be uniformly distributed


within Vp;§ under these conditions, the mean value of t~ can be calculated to be

(~r~rv T i = ~k~]-~z. (24)

In light of the Cylindrical Vp model, the Simple model with fixed tz as described above can be viewed as a "rect- angular" V, model with v, = Vy = 0 and constant v,, as above. Comparisons between the two models are made by choosing an rp (or v,) in Eq. 24 such that r~ for the Cylindrical V, model is equal to the fixed value of t~ assumed in the Simple model--in this manner, the mean signal per atom, [,, will remain the same for both models.

General Outline of Simulations. For either of the models described, the general format of a typical simulation was as follows: the model parameters, T=, Cb, ~,, Xd, were defined. The signal detection limit, X~, was set by the values of Cb, T=, and the desired a. Next the values for Np and t~ were chosen or calculated. Although N, would normally vary for different measurements in an SAD experiment, it was usually fixed at a given value for the computer simulations. The value of t~ was assigned according to the model used; if a Cylindrical Vp model was used, then it was calculated according to Eq. 23 with the position on the y-axis a uniform random variable between - rp and %. Finally, the total signal I t w a s "measured" from a random number generator. A Poisson number generator was used to determine the contribution for each atom separately (with its own value of t~) in the Cylindrical Vp model; when t~ was constant, then a Poisson distribution with a mean given by N,4)st~ + tLb was used to generate the total signal.

One of the advantages of using a computer simulation such as the one above to investigate SAD theory is that the number N. could be fixed for any number of "measurements"--something which may be impossible to achieve in an SAD experiment. This ability was necessary to investigate the behavior of the counting precision, RSDc, at various values of ¢~, Cb, and Np. For observation of the RSD~ for LIF experiments with various values of e~ and Np, it was possible to fix Np and determine Nm from Eq. 19 for many loops; thus, an estimate of a(N~) could be calculated and used in Eq. 20. This value could then be compared to the values predicted from Eqs. 21- 22.

All simulations use arbitrary time units when specifying parameter values for t~, ¢,, etc. (Note: to provide a basis of comparison to past SAD methods based on flowing atoms/molecules through Vp, it has been reported that rr = 10-100 ~s for atomic beam experiments ~6,28 and rr = 1--10 ms for SAD experiments with flowing solu- tions21. 33)

RESULTS AND DISCUSSION

Detection Efficiency at the Intrinsic Noise Limit with LIF. At the intrinsic noise limit, the value of Xd is one single count during Tin. Table III presents the observed detection efficiency, ~d, for various values of ~b,; the Sim- ple model was assumed, with constants ¢, and t r. The

§ T h e condi t ion necessary for th is is t h a t r . <_ ( r . 2 - x . 2 ) '~.

T A B L E I lL Detection efficiency (ed) at the intrinsic limit. The value of ~b, assumes t~ = 20 time units. The values of the expected detection efficiency are taken from Poisson distribution tables. The signal flux has units of count (atom) -~ (time unit) -~.

Sensi t iv i ty

T r u e Observed I t m e a n Detect ion efficiency

(count / Average Var iance a tom) Cs (count) (count) 2 Expec ted Observed

0.1 0.005 0.0853 0.0852 0.0952 0.0816 0.5 0.025 0.5124 0.5149 0.3935 0.3999 1.0 0.050 1.0090 1.0105 0.6321 0.6333 3.0 0.150 3.0093 2.9628 0.9502 0.9528 6.6 0.330 6.6526 6.4640 0.9986 0.9984 9.0 0.450 8.9946 8.8835 0.9999 0.9999

value of Np was fixed at one atom for 8000 measurements; when the signal is at least one count, then the atom is "detected" during Tin.

This simulation represents a test of the Poisson number generator as much as an illustration of SAD at the intrinsic limit. Table III presents the expected and observed values of e~, [~, and at. As expected for a Poisson detection process, the observed mean and variance of the signal due to the single atoms are almost equal for all values of Cs. SAD is achieved at the 95 % confidence level if ~ = 3 counts/atom, and at the 99.86 % confidence level if is = 6.6 counts/atom, as predicted from Table II and Eq. 17. The last row of entries, with a mean signal of nine counts per atom, corresponds to Alkemade's criterion for SAD; this criterion was derived so that S/N = 3 (or, in other words, RSDm = 0.333). The theoretical value of/3 in this situation is 1.234 × 10 -4.

It is important to realize that the detection of individual atoms is possible for all these conditions. Evalu- ation of these techniques in terms of LODs is thus dif- ficult. The value of ed as a measure of the effectiveness of a near-SAD technique is clearly illustrated; this value can be used for the evaluation and comparison of near- SAD methods.

SAD in the Presence of Noise. Once the level of noise during Tm becomes significant, then the value of Xd must increase to keep the probability of false positive at the desired level. The process of detecting single atoms in the presence of noise is illustrated in Table IV for various values of ttb, Cs, and Xd, using the Simple model. All values of X,, Xg, a, and/3 shown in the table are taken from tables of the Poisson distribution. 45 Detection at two different significance levels is illustrated, represented by the two different pairs of values for Xd and Xg for each noise level. In the first row of values for a given gb, a and/3 are chosen to be approximately 5 %, while these probabilities are kept around 0.14% for the second row.

The signal detection limit, X~, for a given noise level is chosen so as to keep the probability of false positives during Tm (denoted by a) below a pro-defined level. How- ever, due to the discrete nature of X,, it is not possible to choose a at any arbitrary level, since a takes on discrete values for different Xd values. The exact values of a for the chosen X~ are shown in the third column in Table IV. The nature of choosing Xd at low levels of noise is clearly indicated: it is rarely possible to choose a to be exactly the value desired (5% or 0.14%); instead, upper toler-


T A B L E IV. Detection and guaranteed detection of atoms in the presence of noise. For each value o f /~ , the first row represents a,/~ -< 0.05 level and the second row gives a,/~ -< 0.0014.

Observed e~

(100 % ) (100 % ) ¢,t~ = ¢,t~ = #b Xd a X~ fl Xd -- #b X~ - #b

1 4 1.90 7.8 4.85 0.5719 0.9558 6 0.06 16 0.14 0.5478 0.9988

2.5 6 4.20 10.5 5.04 0.5555 0.9494 9 0.11 21 0.11 0.5493 0.9993

5 10 3.18 16 4.33 0.5468 0.9550 14 0.07 28 0.13 0.5279 0.9989

10 16 4.87 23 5.20 0.5410 0.9429 22 0.07 39 0.12 0.5230 0.9995

T A B L E V. Comparison of observed and calculated signal variances for the Cylindrical Vp model. The value of 4~, was held constant at 1 count (atom) ' (time unit) '. The units of t, arc arbitrary time units.

Signal To ta l s ignal variance, ~,: Res idence t ime, t~ average (counts) Observed Calcula ted Average Var iance

1.00 1.05 1.08 1.00 0.08 2.53 3.04 3.05 2.51 0.51 4.98 6.93 6.98 4.99 2.01 7.44 11.70 11.95 7.47 4.54

10.09 17.99 18.20 10.00 7.97 15.02 33.49 33.35 15.01 18.32 19.92 51.55 52.00 19.91 32.06 50.37 246.02 246.36 50.26 195.18

ance must be defined, and a chosen to be below these levels. Naturally, similar considerations hold for Xg and fl; the value of Xg is chosen so that fl is as close to the desired value as possible.

When the sensitivity of the technique was such that the signal distribution for single atoms was centered at Xd (S/N ~ 2-3), the observed detection efficiency values at the various levels of noise were those given in column six in Table IV. Due to the asymmetry of the Poisson distribution, the expected value of ~d is slightly higher than 0.5. It is obvious that true SAD, as defined earlier, is not achieved at these sensitivities. On the other hand, when the sensitivity is such that the mean of the distribution of I t is exactly Xg, then the expected value of E, is 1 - /~ , which is the requirement of SAD. Comparison of the values of/3 and the observed ed (column seven of Table IV) shows that this is indeed the case.

"Extra" Variance in the Cylindrical Probe Model. In the discussion thus far, it has been assumed that ~s and ti were both constants for all atoms which enter Vp. How- ever, variance in either of these two parameters results in "extra" variance in the total signal distribution, It. Simulations based on the Cylindrical V v model were specifically formulated to investigate this process by making the residence time, t , a variable depending on the path of the atom through the cylindrically-shaped probe volume. For an LIF experiment in such a situation, the total variance of the signal, at 2, is given by

at 2= it + (Nv¢,)2a2(ti). (25)

A simulation based on the Cylindrical Vp model at the intrinsic limit (simple integration of signal over Tm) was run 8000 times for several values of Tr (t~ = t~ for cw LIF) and Cs = 1 count (atom) -~ (time unit)-L Table V shows the results of the simulations. The observed values st 2 and s2(t~) are given in the table; the value of s2(tr) was used as an estimate for a2(t~) in Eq. 25 to calculate the expected total signal variance. The calculated values of at 2 compare very well to the observed values for all simulations. There was good agreement for simulations with varying levels of Np and ¢~ as well (although the results are not shown here).

In Table V, if ti were constant for all atoms entering Vp, then of course the signal variance would be approximately equal to the mean; the larger signal variance due to variation in t~ can be clearly seen in Fig. 7 for 8000 loops of the simulation with values of is = 12.35 counts/

atom and N v = 1 atom. The observed distribution of Is (intrinsic case) and the Poisson distribution which would be expected with fixed tr are shown in the figure.

The consequences of variable ti can be easily understood from Eq. 25 in light of the discussion of e, and RSDc thus far. Obviously, an increased variance would decrease both E~ and RSDc, the latter particularly since the total signal variance is proportional to Np 2. A sensitivity high enough to result in a true SAD technique for constant ti would perhaps not be high enough if the value of a2(ti) were large; in addition, since ~t 2 is proportional to Cs 2, it might be that a rather large improvement in sensitivity would be necessary to achieve SAD. A more effective method of achieving SAD might be to decrease the "extra" variance term by decreasing a2(tl), even at a cost to S/N. Finally, the value of e~ as an FOM for near-SAD methods is even more clear, since S/N does not reflect this "extra" variance. For example, the two distributions in Fig. 7 each possess the same S/N ratio; if the noise level were such that Xd = 5 counts, then the detection efficiency of the Poisson-distributed It would be significantly higher than that due to the Cylindrical Vp model.

Counting Atoms at the Intrinsic Limit with LIF. The difference between the signal measurement precision, RSDm, and the counting precision, RSDc, when a given

'~ 2 0 0 0 ,o

0

0

v

O

1 0 0 0

0 0

(~ )T I = 1 2 . 3 5

5 10 15 2 0 25 3 0

Signal (counts)

• O b s e r v e d - - P o i s s o n

FIG. 7. Compar i son of the observed signal d i s t r ibu t ion I, wi th Np = 1 (8000 m e a s u r e m e n t s ) for the Cylindrical Probe mode l with the Pois- son d i s t r ibu t ion which would be expected for the same S /N level for t he Simple model.


TABLE VI. Counting precision, RSD~, at the intrinsic limit: comparison of theoretical and observed values.

Sensitivity (counts/atom)

6.6 34 35 100

RSD¢ RSD~ RSDc RSD,

N~ RSD~ Obs Calc RSD~ Obs Calc RSD~ Obs Calc RSD~ Obs Calc

1 0.389 0.339 0.366 0.171 0.054 0.049 0.169 0.051 0.052 0.100 0.000 0.000 2 0.275 0.293 0.294 0.121 0.099 0.099 0.120 0.095 0.095 0.071 0.012 0.010 3 0.225 0.238 0.238 0.099 0.102 0.101 0.098 0.099 0.099 0.058 0.019 0.021 4 0.195 0.211 0.212 0.086 0.096 0.095 0.085 0.093 0.093 0.050 0.026 0.028 5 0.174 0.184 0.185 0.077 0.088 0.088 0.076 0.086 0.086 0.045 0.033 0.032

10 0.123 0.127 0.128 0.054 0.061 0.061 0.053 0.061 0.060 0.032 0.034 0.034 15 0.101 0.103 0.103 0.044 0.048 0.048 0.044 0.047 0.048 0.026 0.030 0.030 20 0.087 0.089 0.089 0.038 0.041 0.041 0.038 0.040 0.040 0.022 0.026 0.026

number of atoms pass through V~ during a measurement, was illustrated earlier in Figs. 4 and 5. Calculation of the theoretical value of RSDc, for fixed values of Np, {s, and #b, is represented by Eqs. 19-22. The theoretical values are computed in the case of Poisson distribution of I, with a simple computer program, since the manual com- putation can be rather tedious. In particular, the minimum value of the sensitivity, (/s)~, necessary to achieve a certain pre-defined counting precision, is of interest as a comparison to the sensitivity necessary for SAD.

A computer simulation can be used to compare the observed value of RSD¢ with the theoretically calculated values, since in a computer simulation the number of atoms probed can be held constant for any number of measurements. In the simulations, the measured number of atoms, Nm, is computed for each measurement according to Eq. 19; the standard deviation of Nm is then used to calculate the observed value of RSDc and comparisons to the theoretical value can be made. Table VI shows the results of 8000 measurements at the intrinsic limit for various values of ~, and Np, along with comparisons to the theoretical values for both RSD~ and RSD~.

A number of interesting observations can be made from Table VI. First, in all cases the observed and calculated values of RSDc agreed very well. The column with a sensitivity of 6.6 counts/atom represents the requirement of SAD at the 99.86 % confidence level; clearly, RSDc is higher than 0.10 (the desired precision) for Np < 16 atoms. This observation illustrates the fact that a sensitivity which is necessary to achieve SAD may not be sufficient to count any number of atoms with precision; at lower analyte concentrations, the counting precision is quite high.

Another interesting observation involves the difference between RSD~ and RSDc. As predicted by Eq. 18b, the value of RSDm decreases as Np increases. However, the values of RSD~ at low numbers of probed atoms are markedly different from the expected RSD~ and increase to a maximum value before once again decreasing with increasing Np. From both the calculated values of RSD~ and the simulations, the value of ([~)~ necessary for RSDo -< 0.10 is approximately 35 counts/atom at the intrinsic limit. This value is 5.3 times the necessary sensitivity for true SAD (at the 0.0014 significance level).

Finally, the comparison of RSDm and RSD~ at 100 counts/atom is interesting since this is the sensitivity predicted for precise counting on the basis of RSD~ only. As was illustrated in Fig. 4, the counting precision at this

sensitivity is much greater than predicted by RSDm for small numbers or atoms, since the overlap between distributions of different values of Np is slight.

Counting Atoms with LIF in the Presence of Noise. The presence of increasing levels of background noise during Tm will tend to broaden the signal distributions, It, for fixed values of Np; hence, it is expected that RSDc will increase as well for a given sensitivity at the analyte concentration level. Table VII reflects the increased signal variance in increased levels of ({s)c necessary to achieve precision counting at the same levels of/~b as shown in Table IV. It is interesting to note, through comparison of Tables IV and VII, that the value of ({s)c does not increase nearly as quickly as the minimum sensitivity required for SAD as ~b increases above the intrinsic- limited case.

CONCLUSIONS

This work has introduced some general principles which may apply in many ultratrace laser-based experiments which possess sensitivity high enough to measure the signal associated with single atoms in the laser beam. The detection of atoms in the presence of background noise, the use of the detection efficiency as a means of defining, evaluating, and comparing (near-)SAD methods, the presence of "extra" variance due to variation in (~s o r ti, and the extra sensitivity required to count atoms by LIF have all been discussed and demonstrated with the computer simulations presented here.

However, the SAD theory presented, along with the computer simulations used, is only as good as the model upon which it is based. A brief look at some of the assumptions inherent in the general SAD model presented here is warranted. The assumptions of Poisson processes

TABLE VII. Counting precision in the presence of noise. The second column gives the minimum sensitivity (connts/atom)--both theoretical and observed values--needed to precisely count atoms at all levels. The value of RSD c and the corresponding Np represent the most imprecise counting levels.

(i,)c RSDc #b (counts/ Np

(counts) atom) (atoms) Calc'd Observed

1 35 3 0.0995 0.1001 2.5 36 3 0.0982 0.0985 5 37 3 0.0976 0.0991

10 39 2 0.0969 0.0940


for both the background noise and the detection process are not serious obstacles. It is quite likely that in many situations the first assumption is not valid; however, some assumption of the form of Ib had to be made, and a distribution other than a Poisson merely signifies that a different value of X , must be chosen. The assumption of a Poisson detection process is usually a valid assumption in the time frame of most experiments. However, the characteristics of Poisson processes must be kept firmly in mind. The presence of metastable traps, detector "dead" time, or space-charge effects can all alter the distribution of the detection process.

One of the largest idealizations in the general SAD model involves the use of a well-defined probe volume. In many analytical experiments, it is simply not possible to have a well-defined probe volume due to either the collection/detection efficiency of laser-induced events in the atoms or the spatial characteristics of the laser beam itself. For example, a common method of describing the radius of a laser beam involves the use of the 1/e distance--i.e., the distance at which the intensity decays to 1/e of its peak value. However, the laser beam obviously still has some intensity at values beyond this distance, and may cause atoms to be detected with a nonzero probability. Likewise, in LIF experiments, the collection efficiency of most optical systems is a function of the distance of the emitter from the central axis of the system. Stating where the probe volume "stops" may not have much meaning in LIF experiments, although the use of spatial filtering can help in this.

The concept of the detection efficiency itself is obviously only as well defined as the probe volume, despite the fact that the requirement for a method to be considered SAD has been given in terms of this FOM. Thus, it may well be that the concept of an SAD experiment may be an idealization if there is an attempt to apply SAD theory as presented to a particular experiment.

Despite some of the sobering realities mentioned here, many of the concepts of the theory of SAD as presented are of value to the general laser spectroscopist. Even the analytical chemist interested only in ultratrace analysis by laser-based techniques should probably be aware of many of the concepts presented for SAD, even if there is no desire to count individual atoms, since at the detection limit there may be few atoms or molecules of analyte being actively probed by the laser. Obviously, the analytical or physical chemists, who are actually concerned with measuring signals or properties dealing with single atoms, struggle with concerns which do not arise in more conventional experiments; the formulation of the general SAD model and the related concepts have demonstrated this fact most clearly.

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Estimating Detection Limits in Ultratrace Analysis. Part II: Detecting and Counting Atoms and Molecules