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Anal. Chem. 1980, 52,
eliminates both the cleanup and the chromatography. In addition, instrument time is reduced to about 10-15 min/ sample. In general the analytical utility of the combined TSQ and CAD method will be greatest for those problems which require detection and/or quantitation of specific chemicals in complex environmental or biological matrices. Several known compounds, perhaps as many as 30 to 50, can be determined in a single sample under ideal conditions by making use of slow heating regimes and fractional vaporization of the sample into the ion source along with computer controlled selected ion monitoring in both Q1 and Q3. Rapid screening of a large number of samples for a few specific compounds is a chore t h a t seems ideally suited for the TSQ/CAD approach. Wet chemical fractionation followed by conventional GC/ MS computer techniques will remain the method of choice for identifying all components in a particular mixture. Combined GC/TSQ/CAD should be of particular utility for low level quantitation of biologicals since the TSQ eliminates both noise due to GC column bleed and the possibility that fragment ions from high molecular weight impurities will interfere with accurate measurement of ion abundances during quantitation of lower mass species. The end result is increased sensitivity and improved confidence in the data obtained during quantitation studies.
ACKNOWLEDGMENT T h e authors are indebted to M. Story, G. C. Stafford, and W. Fies of Finnigan Corporation for many helpful discussions during the course of this research.
390-394
LITERATURE CITED Kondrat, R . W.; Cooks, R. G. Anal. Chem. 1978, 50, 81A-92A. Bente, P. F., 111; McLafferty, F. W. "Mass Spectrometry"; Merritt. C., McEwen, C. N.. Eds.; Marcel Dekker: New York, 1979; Chapter 3. Youssefi, M.; Cooks, R. G.; McLaughlin, J. L. J. Am. Chem. Soc.1979, 101, 3400-3402. Kruger, T. L.; Kondrat, R. W.; Joseph, K. T.; Cooks, R. G. Anal. Biochem. 1979, 96, 104-112. Levsen. K.; Schuken, H. R. Biomed. Mass Spectrom. 1978, 3 , 137-139. Maquestiau, A.; Van Haverbeke, Y.; Fhrnmang, R.; Miiprewe, H.; Kaisin, M.; Braekman, J. C.; Daioze, D.; Tursch, B. Steroids 1978, 31, 31-48. Yost, R. A.; Enke. C. G. J. Am. Chem. SOC. 1978. 100, 2274-2275. Beynon, J. H.; Brothers, D. R.; Cooks, R . G. Anal. Chem. 1974, 46, 1299-1302. Yost, R. A.; Enke. C. G.; McGilvery. E.; Smith, D.; Morrison, J. D. Int. J. Mass Spectrom. Ion Phys. 1979, 3 0 , 127-136. Hunt, D. F.; Stafford, G. C.; Crow, F. W.; Russell, J. W. Anal. Chem. 1976, 48, 2098-2105. Srnit, A. L. C.; Field, F. H. J. Am. Chem. SOC. 1977, 99, 6471-6483. Hunt, D. F.; Crow, F. W. Anal. Chem. 1978, 5 0 , 1781-1784. Rosenstock, H. M.; Draxl, K.; Steiner, 6. W.; Herron, J. T. J. Phys. Chem. Ref. Data 1977. 6 , Suppl. 1. 774-783. Bartrness, J. E.; McIver, R. T. "Gas Phase Ion Chemistry"; Bowers, M. T., Ed.; Academic Press: New York, 1979; Chapter 11. Benson, S. W. "Thermochemical Kinetics", 2nd ed.;Wiley Interscience: New York, 1976. McClusky, G. A.; Kondrat, R . W.; Cooks, R. G. J . Am. Chem. SOC. 1978, 100, 6045-6051. Yamaoka. H.; Pham, D.; Durup, J. J. Chem. Phys. 1969, 57. 3465-3476.
RECEIVED for review October 22,1979. Accepted December 6, 1979. A preliminary account of this work was presented a t the 27th Annual Conference on Mass Spectrometry and Allied Topics, Seattle, Wash., June 1979, Paper TPMPG. This research was supported by grants from the U.S. Army Research Office, Grant No. DAAG 29-76-G-0326, and the U.S. Environmental Protection Agency, Grant No. R805790-01.
Simultaneous Determination of 20 Trace Elements in Geologic Samples by the Isotope Dilution Method Combined with Spark Source Mass Spectrography '
Hans-Josef Knab" and Heinrich Hintenberger Max-Planck-Institut
fur Chemie
(Otto-Hahn-Institut),
0-6500Mainz, West Germany
A technique is described to simultaneously determine 20 or more trace elements in geologic samples applying the isotope dilution method combined with spark source mass spectrography. The advantages of this technique are: (a) the measurements are independent from calibration standards and (b) the analyses of sundry samples with similar chemical compositions are simplified by using isotopically spiked graphlte. Concentrations determined by this method are in good agreement with those obtained by other analytical methods. The average of all relative standard deviations is better than 10%.
During the past years many studies have been concerned with the calibration of the spark source mass spectroscope. In most of them (1-3) relative sensitivity factors were used Present address: Institut fur Kristallographie und Petrographie, ETH Zurich, CH-8006 Zurich, Switzerland.
to calibrate the measurements. This procedure, however, is limited by the availability of well analyzed standard samples. Also, the results of the concentration measurements on many standard samples, obtained by different analytical methods, exhibit large variations. It is obvious that such uncertainties have unfavorable influences on the accuracy of the final results. In some other works (4-7) the isotope dilution method has been employed. Most of these studies (4-6)involved simultaneous determinations of only a few elements in different metal samples. In these studies the enriched isotopes and the samples were dissolved together and, after drying, the isotope ratios in the mixtures were measured by mass spectrography. T h e determined concentrations were in good agreement with the specified values in the test samples. Jaworski and Morrison (7)successfully analyzed geological samples utilizing the multielement isotope dilution technique. They added the enriched isotopes to the graphite which is necessary to increase the electrical conductivity of the sample to be analyzed. Then they mixed this spiked graphite with the USGS standard diabase W1 and measured the isotopic
0003-2700/80/0352-0390$01.00/0@ 1980 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 52, NO. 3, MARCH 1980
Table I. Trace Elemental Concentrations in the Holbrook Meteorite (in ppm by weight) concentrations determined by isotope dilution method cu Sr
Te
Ba Sm
Dy
( a ) 86.4
6.9 ( b ) 87.3 c 7 . 0 ( a ) 11.7 i 0.9 ( b ) 12.2 t 1.0 ( a ) 0.40 5 0.02 ( b ) 0.42 z 0.02 ( a ) 4.87 I0.39 ( b ) 5.40 t 0 . 4 3 ( a ) 0.29 t 0.02 ( b ) 0.29 I0 . 0 2 ( a ) 0.36 k 0.03 ( b ) 0 . 3 2 i. 0.03 t
Allende sensitivity f ac torsu 87 13
i t
20 2
1 . 2 i 0.3
range of literature values 74-124 9-14.3 0.26-1.94
7.7
t
1.0
3.2-12.2
0.41
t
0.08
0.22-0.28
0.66
*
0.11
0.36-0.46
Ref. 11. ratios of the spiked elements in the mixture. Knowing the altered isotopic ratios, the natural ratios, and the elemental concentrations in the W1 sample, they were able to determine t h e added amounts of the enriched isotopes. With this standardized spiked graphite, they subsequently determined t h e concentrations of nine elements in the USGS standard basalt BCR-1. For the nine measured elements, they obtained a n average RSD of 5.4% and an average error of 5.5%. This procedure simplifies the standardization of the spiked graphite, which can be used for many analyses. Furthermore the spike concentrations are determined in the final graphite. T h u s uncertainties caused by a partial loss of the enriched isotopes during the handling process cannot arise. But there are two disadvantages in this procedure. First, standard samples of suitable composition to standardize a multiisotopic spiked graphite are missing in most cases. Second, Flanagan's recommended values for the W1 standard (8),which were used by Jaworski and Morrison (7) for the standardization of their spiked graphite, have inherent errors. W l data compilations by Fleischer (9) indicate t h a t nearly all published concentrations of the different elements vary by a factor of 2. Although most data lie within a range of about 1070,the reiiability of that kind of calibration has in principle an inherent error of a few percent, which has to be added to t h e results. T o avoid such problems, we standardized the spiked graphite in the way reported by Donohue e t al. (IO). Precisely determined quantities (milligrams) of the enriched isotopes were dissolved. Then t h e spike concentrations were diluted by adding water. Thereupon the appropriate amounts of these spike solutions were mixed with the graphite.
EXPERIMENTAL Procedure. The normal approach in the isotope dilution m.ethod is to dissolve the sample and the enriched isotopes t,ogether or in the same way and then to make the mass spectroscopic measurements. We discovered that this method is not practicable in spark source mass spectrography, especially in measuring trace elements. This is because too many molecules of the sample's main elements overlap the mass spectra ofthe trace elements. Since the formation of molecules is essentially less when the sample is sparked in the undissolved state. we only dissolved the enriched isotopes and added them to the undissolved sample. Two different techniques are suitable for this: the addition of the isotope solutions (a) directly to the sample or (b) to the graphite. Technique b has two advmtages relative to technique a. These are, first. that, the whole standardization procedure can be done wit,h the graphite. If any contamination is generated, which is important in the case of rare and precious samples, only the graphite would be contaminated and not t.he sample. Second, if a set of similar samples (in our case nine carbonaceous chondrifea, a ~pecialkind of rare meteorites) has to be analyzed. the
391
Table 11. The 20 Trace Elements Analyzed enriched isotope 6jcu
"Ga 73Ge "Sr 91Zr
I19Sn 12ISb '25Te '=Ba 143Nd
chemical compound CUO Ga,O, GeO, Sr(NO,),
ZrO SnO Sb
Te Ba( NO, 1 2 Nd203
147Sm
Sm,O,
151Eu
Eu203
157Gd
Gd,O, DY@,
I6*Dy 171 Yb 179Hf ia3w '''Re l"Pt 205pb
Yb,O, HfO,
wo
1
Re Pt Pb(N0
solvent HC1 HF NaOH HCI H2SO4 H2SO4 HC1 HCl HC1 HC1 HC1 HC1 HC1 HC1 HC1 H,SO, NaOH HNO, HC1 + HNO, HC1
tedious standardization procedure need be done only once. To compare the two methods, we determined the concentrations of six trace elements in two aliquota of the Holbrook meteorite using these two techniques. The results are shown in Table I under (a) and (b). For comparison, the concentrations determined by using sensitivity factors, which were obtained from the Allende meteorite (II), and the literature values are included. Both isotope dilution measurements are in good agreement within the standard deviation and are also consistent with the literature values. Furthermore, it is obvious that the isotope dilution measurements have standard deviations (mean RSD 7.4%) which are distinctly less than those of the results obtamed by the calibration with the sensitivity factors (mean RSD 19%). The lower standard deviations yielded by the isotope dilution method are not unexpected. the precision of the final results is mainly given by the precision of the isotope abundance measurements By using sensitivity factors, however. the precision of the final results consists of (a) the standard deviation of the sample measurement, (b) the standard deviation of the measurement of the standard sample for the determination of the sensitivity factors, and (c) an error. caused by the not exactly known concentrakons in the standard sample For our further measurements %+emade use of technique b Selection of the Measurable Elements. Some photoplates from each sample were exposed before spilung Elements selected for later quantitative determination were those which had at least two interference-free isotopes on the basis of the mass spectra evaluation. The criterion for interference-free isotopes was an ageement of the measured w t h the natural isotopic abundmces better than 5 % . In this respect we were able to measure quantitatirely 20 trace elements with an atomic number 1 2 9 (Cu)in the rneteoritx samples (see Table 11) It should be added that we were interested only in memuring trace elements in this study. The enriched :sotopes were obtained from the Oak Ridge National Laborator). Oak Ridge, Tenn E n x h e d isot3pe~of one fdrther element. Os were not avcilable at th: time of order. Preparation of the Spiked Graphite Midigram quantities of the enriched isotopes w r e dissolved either in double distilled deionized H 2 0or in pureJt reagents IhIerck-Suprapur, gee Table Ii) One stock solution was produced fcr each :sotope. To FTsvent admrptlon effects, these stock 4ut:oris had spike concentraticm nf about 1000 ppm by weight and s e r e stored in po2yeth)lene bottles. Smdl mounts of each stock eoluticr, were frrrthe: &luted hy adding H,O untd the desired cGncent:atioZlS of the enriched :sotopes were obtained. To establish thnse ecncentrat:ons ts:n different facta ha\e to he considered First, the :Sotopic r a t m in the mixture of the spike and the i;unple &odd be c!(w trr one, the most favorable ones to be measured from photographic plates That can be estimated by the evaluation of the rinspiked mass spectra. Second, the amounts of thL enrl-hed 1 s r r i ~ ~added w ~ to the wniple miib+ he
392
ANALYTICAL CHEMISTRY, VOL. 52, NO. 3, MARCH 1980
Table 111. Blank Measurements for Impurities in the Final Spiked Graphite (ppb by weight) cu Ga Ge
Sr Ba
5 0' 2 00
w
100
Ofll
15
10 5
* These are probably molecules. (See text). propagation small. It is shown (12) that the standard deviation caused by the isotope abundance measurements has its lowest propagation to the final results if the isotopic ratio in the mixture is the geometrical average of those in the spike and in the sample. Since these two facts are usually not fulfilled at the same time, we added the spikes in such quantities that the final isotope ratios were between 1 and the geometrical average. In no case did the added spikes differ by a factor greater than 3 from these values. Anyhow, in such a way estimated amounts of the diluted stock solutions were weighed exactly and added to the graphite as well as approximately 1000 ppm by weight silver dissolved in HN03 to be used later to construct the transparency curve. Some drops of ethanol were used to reduce the surface tension and some drops of NH40H to neutralize the remaining acid. Then the graphite was dried in a Teflon beaker at a temperature not higher than 70 O C . During the drying process the suspension was frequently stirred to prevent the effects of settling. Possible H 2 0 residues were evaporated in a drying chamber under vacuum. Blank Measurements. To check whether contaminations disturb the results, several purity measurements were made. The pure graphite, the spiked graphite, the enriched isotopes, and all solvents used including the ethanol and the distilled water were analyzed with the mass spectrograph. The observed impurities in the final spiked graphite of those elements which were measured in this study are summarized in Table 111. They are either from the graphite (W) or from the solvents (Sr, Ba). Impurities which could have had disturbing effects on the analysis were not detected in the enriched isotopes. This means that possible contaminations in the quantities of the enriched isotopes used for analysis are, if present a t all, lower than the detection limits. Non-natural isotope ratios of Cu, Ga, and Ge indicate that the apparent contaminations of these elements are probably simulated by non-identified molecules. The element Mo existed in the graphite used in a weight concentration of about 3 ppm which prevented us from measuring this element. Further impurities were not detected. Mass Spectrographic Measurements. The spiked graphite was added to the sample (1 part by weight graphite to 2 parts by weight sample) and mixed for 0.5 h in an agate mill using agate balls. Then it was pressed into electrodes and analyzed in the conventional manner (13). Ten photoplates (Illford Q2) from each sample were exposed using a double focusing mass spectrograph of the Type MS 702 from AEI, Manchester, Great Britain. Each photoplate had 15 graded exposures: 6 plates with maximal 50 X C to determine the elements with weight concentrations of more than 5 ppm and 4 plates with exposures up to 2 X lo4 C to determine the elements with lower concentrations. Evaluation of t h e Photoplates a n d Calculations. The photoplates were evaluated with a Steinheil photometer. As mentioned previously, about 1000 ppm silver was added to the spiked graphite. The transparency curve as a function of the exposures was measured using the line transparencies of the silver mass spectra. The exposures were graded such that on each photoplate 16-20 Ag lines were measurable outside the saturation transparency. These points were sufficient to fit the transparency curve based on the formula of Franzen et al. (14) and to determine the characteristic curve parameters. The altered isotope ratios of the spiked elements were then determined with these parameters in each exposure separately. The element concentrations were computed from Equation 1: bi -
Cbk
X = YM,, - z Cak - a, where X is the concentration in ppm by weight, Y is the total weight in fig of the enriched isotope added to the whole spiked graphite, M,, is the ratio of the natural atomic weight to the atomic
I
2 01
/
1':,,, I
l
l
l
I
l
I
I
,
OLr dato w#th mecn d e v i c t l o n s I
I
I
I
I
I
01
I
,
I
I
/
I
I
l
10
l I/)
PP"'
1GG
Figure 1. The Allende meteorite: a comparison of our data with literature values
Table IV. Sn and Sb Concentrations in Two Meteorite Samples (in ppm by weight) element
sample (meteorite)
our data
literature data (range)
Sn
Murray Orgueil
0.91 3 0.10 0.16 * 0.02
0.89-1.05 0.13-0.19
Sb
weight of the spike, aiand ak are the natural abundances of the analyzed isotopes i and k, bi and bk are the abundances of the isotopes i and k in the spike, C is the mass spectrographically measured altered ratio of the isotope i to k in the mixture of the spiked graphite and the sample, and z is a conversion factor which is given by Equation 2: 2
=
ws,/s,w,
(2)
where W,, is the weight of the spiked graphite added to the sample, W , is the weight of the sample, and W is the weight of the totally produced spiked graphite (all weights in grams).
RESULTS AND DISCUSSION T h e accuracy of the described method is demonstrated in Figure 1. In this figure, our determined concentrations in the Allende meteorite are compared with all available literature data. Allende was taken for this comparison because there exist many analyses of i t and, furthermore, its composition is very similar t o t h e samples analyzed in this study. T h e literature values in Figure 1 appear with the ranges of all available data; our results are shown with t h e standard deviation bars (la). Data from this study fall in t h e ranges of the concentrations measured by different authors for nearly all elements. Only the Sn and Sb data lie off the straight line signifying agreement. T h e two disagreeing points may be caused by inhomogeneity or contamination of t h e sample analyzed. It cannot be attributed to erronous measurements. The latter conclusion is confirmed by the results of some other samples which were analyzed using the same spiked graphite (see Table IV). Our determined concentrations of S n and
ANALYTICAL CHEMISTRY, VOL. 52, NO. 3, MARCH 1980
393
Flgure 2. W1: a comparison of the results obtained with published data. Published data are arranged by analytical methods. 'Value recommended by Flanagan (8). bRef. 9
Table V. Concentrations in the USGS Standard Dibase W 1 (in ppm by weight)
element
cu Ga
Ge Sr Zr Sn Sb Te
Ba Sm Nd Eu Gd
2
Hf Pb a
our results
11oi 9 19.1 I 1.9 1 . 7 8 i 0.16 201 f 1 9 88.9 t 12.7 2.80 I0.32 1.15 i 0.17 0.10 i 0 . 0 1 1 6 9 i 15 4.00 i 0.34 15.9 i 1.1 1.18 I 0.11 4.00 t 0.26 3.91 5 0.29 2.16 t 0 . 1 3 1 . 9 6 i 0.16 6.47 L 0.67
recommended values by Flanagana 110 16 1.3 190 105 3.2 1.0 1 160 3.6 15 1.11 4
4 2.1 2.67 7.8
Ref. 8 .
S b in these samples are in good agreement with some quoted literature data. T h e concentrations in the nine meteorites determined in this study were previously reported (15). The average of all relative standard deviations (of 20 elements in 9 samples) is about 10%. In comparison with the precisions yielded when using sensitivity factors (11) the present analyses are improved by a t least a factor of 2. The remaining deviation of nearly 10% is primarily caused by the inhomogeneity of the electrodes but also by the evaluation of the photoplates. The analysis of a further standard sample, the USGS standard diabase W1, provides an additional test of the reliability of our method. Because its elemental concentrations are distinctly different from those in the meteorites, another spiked graphite had to be produced. The results of our analysis together with the recommended values of Flanagan (8) are summarized in Table V. A comparison of the present results with those obtained by other analytical methods is shown in Figure 2. In this figure, our results are indicated
by big horizontal bars. The vertical bars indicate from the right to the left the ranges of NAA, MS, and XRF data. The total ranges of all published data collected by Fleischer (9) are given by the bars furthest to the right. The recommended values of Flanagan (8) appear as crosses. There is good agreement for all elements except Te. For the latter element, only an upper limit is given by Fleischer (9). The elements W, Re, and Pt, which are determined in the meteoritic samples, could not be measured in W1. Their concentrations are too low (.=Ippb) to be detected with photographic plates. Also the oxides of the very abundant rare earth elements interfere with the mass spectra of these ultra trace elements, so that an evaluation seems to be impossible. In an earlier section, the amount of 20 elements is mentioned to be measurable using the described technique. This amount of 20 elements relates only to the trace elements in the samples analyzed in this study: the carbonaceous chondrites. In principle, elements like Zn, Se, Ag, Ru, Pd, and Cd, whose mass spectra are disturbed by molecules in these samples; or Mo, if a cleaner graphite is used, or the main and minor elements Mg, Si, S, C1, K, Ca, Ti, Cr, Fe, and Ni which were not included in this study, should also be measurable in geologic samples of different compositions using the described isotope dilution technique. The two monoisotopic elements I and Th, as well as U should likewise be determinable in the same way when spiking the radioactive isotopes lz9I,230Th, and 235U.
LITERATURE CITED (1) Hintenberger, H.; Jochum, K. P.; Seufert, M. In "Analyse extraterrestrischen Materials", Kiesl, W., Malissa, H., Jr.; Springer Verlag: WienNew York, 1974; pp 125-145. (2) Jaworski, J. F.; Morrison, G. H. Anal. Chem. 1974, 46,2080-2084. (3) Taylor, S. R . ; Gorton, M. P. Geochim. Cosmochim. Acta 1977, 4 7 , 1375-1380. (4) Leipziger, D. F. Anal. Chem. 1085, 37, 171-172. (5) Alvarez, R.; Paulsen, P. J.; Kelleher, D. E. Anal. Chem. 1969, 47, 955-958. (6) Paulsen, P. J.; Alvarez. R . ; Mueller, C. W. J . Appi. Spectrosc. 1978. 30,42-46. (7)Jaworski, J. F.; Morrison, G. H. Anal. Chem. 1975, 47, 1173-1175. (8)Flanagan, F. J. Geochim. Cosmochim. Acta 1973, 37, 1189-1200. (9) Fleischer. M. Geochim. Cosmochim. Acta 1989, 33,85-79. (10) Donohue, D. L.; Carter, J. L.; Franklin, J. C. Anal. Lett. 1977, 70, 371-379.
Anal. Chem. 1980, 52, 394-398
394
(11) Knab, H. J. "Die Bestimmung von 60 Haupt- und Spurenelementen in den Meteoriten Dimmitt, Gruver und Mills durch Funkenmassenspektrographie", Diplomarbeit, Max-Planck-Institut fur Chemie, Mainz,
(15) Knab, H. J.; Hintenberger. H. Meteoritics 1978, 73, 522-527.
1975.
(12) Riepe, W.; Kaiser, H. Fresenius' Z.Anal. Chem. 1966, 223, 321-335. (13) Hintenberger, H. Fortschr. Mineral. 1977, 5 4 , 141-166. (14) Franzen, J.; Maurer, K. H.; Schuy, K. D. z. Naturforsch. 1966, 21a, 38-62.
RECEIVED for review August 27, 1979. Accepted November 29, 1979.
Calculation of Elemental Compositions from High Resolution Mass Spectral Data R. Geoff Dromey" and Gordon T. Foyster Department of Computing Science and Computer Cenfre, University of Wollongong, P.O. Box 1144, Wollongong, N.S. W. 2500, Australia
The calculation of elemental compositions for high resolution mass spectral data can be structured in such a way as to mlnhlre the number of steps to generate each new candidate. When this Is done considerable gains in efficiency are achleved. Furthermore, If metkulous care Is taken to examine only those regions of the compositlon search space that can posslbly lead to valid compositions, then even much larger gains in computational efflciency are made. Both these aspects of the elemental compositions calculation are explored in detall. The outcome has been the development of a new algorithm for elemental composltlon calculations that is approxknately 100 times faster than currently available algorlthms for typical spectra. The new algorithm alleviates the problem of excessive computation times for both high mass values and for situations where six or more element types must be considered.
The assignment of possible elemental compositions to peaks in a high resolution mass spectrum has proved to be a formidable computation even for modern high-speed computers. The combinatorial nature of the problem almost invariably makes the calculation a time-consuming task. The development of computerized high resolution mass spectrometer (HRMS) systems has resulted in a greatly enhanced capacity for generating high resolution data. This is particularly so for GC-HRMS-computer systems. These developments together with the increasing employment of high resolution mass spectrometry in analytical applications have served to underline the need for very efficient computation of elemental compositions. A number of computer algorithms (1-3) and methods ( 4 , 5 ) have been suggested for the computation of elemental compositions. These algorithms perform satisfactorily when 0003-2700/80/0352-0394$01 .OO/O
only 3 or 4 element types are considered. Beyond this they are usually unacceptably slow. Robertson and Hamming ( I ) in their treatment of the problem have recognized the difficulty of performing these computations rapidly for other than a limited set of elements with very restricted ranges. By use of a relatively sophisticated backtrack programming technique they were able to develop an algorithm that was significantly more efficient and more widely applicable than earlier schemes. In the present work it will be shown how to make close to a further 100-fold gain in computational efficiency. B A S I S OF A L G O R I T H M T o place the current work into perspective, the problem shall be defined more explicitly and it will be shown how earlier algorithms were used to solve it. In computing the elemental composition for any particular mass what must be done is first to decide on what elements may be present in the composition. The problem must then be further constrained by setting limits on the number of atoms of each type that could realistically (chemically) be present in the composition (e.g., the upper limits might be C2,, H,,, Og, N8, Cl,, SJ. The task then is to find all combinations of elements from these ranges that add up to the accurate mass being considered (that is, to within some predefined error tolerance, perhaps 5 ppm). In the above example this amounts to 24 X 50 X 8 X 8 X 4 X 4 or 122880 possible combinations that need to be examined. In any given problem only a minute fraction of these combinations is successful in satisfying the constraints imposed by the accurate mass and its associated error tolerance. Earlier algorithms for solving this problem did so by essentially generating all possible combinations (perhaps with the exception of carbon which was factored out) and then checking each combination against the accurate mass. The reason such an approach is so slow is because of the large number of candidates that are generated, many of which have no chance 1980 American Chemical Society
