quires an electrical detection system which was unavailable during the course of this investigation. S p a r k Source Mass Spectra of Mixtures. An important unanswered question concerns the processes in the rf spark discharge plasma which led to the observed molecular ions. There are three possible explanations for the observation of ions with masses greater than the molecular weight of the carboxylate under investigation. The first model is the evaporation of the carboxylate as a polymer, (MA)n, where M is again the metal and A the carboxylate, followed by ionization and very rapid dissociation to the MzA+ species and subsequent rearrangements and fragmentations. This model is consistent with the behavior of inorganic compounds in high-temperature mass spectrometers equipped with a Knudsen cell. However, the heavy amount of sputtering in the ion source region of the mass spectrometer suggests that the immediate region of spark discharge is a moderate-to-high-pressure source of ions. Thus, the possibility that an important source of the observed molecular ions is ion-molecule reactions within the rf discharge must be considered as a second model. A third model postulates the formation of observed species by solid-state processes followed by evaporation and ionization. To gain some insight into these possibilities, a study of the spark source mass spectra of selected mixtures was undertaken. Mixtures containing approximately equimolar amounts of sodium butyrate and either copper metal, CuC12, or KC1 were subjected to sparking. Sodium butyrate was chosen because it gives an intense MzA+ ion and the spark source mass spectra of copper(I1) butyrate and potassium butyrate were known. The CuClz was chosen as the second component since Cu(I1) has a great tendency to form chelate complexes. The use of copper metal then should allow the distinction between processes occurring within the electrode and processes occurring in the plasma phase involving CUI+ or Cu2+ ions. This assumption requires that metal ions in the plasma phase have the same electronic configurations regardless of their different chemical sources and is probably valid when consideration is given the nearly equal sensitivity of the spark source
mass spectrometer to the various elements, regardless of chemical origins (38). Finally, KCl was chosen as the second component of the mixture because potassium does not show a great tendency to form chelate complexes in solution chemistry. Data were recorded from exposed plates for the metal-containing molecular ions prominent in the mass spectrum of each pure carboxylate. The ions chosen were M O + , MzO+, MA+, MzA+, and NaMA+, where M is either copper or potassium and A is the butyrate group. The Na&H702+ (MzA+) peak from sodium butyrate .at m / e 133 was used to compare relative exposures. These data are summarized in Table IV. None of the predicted “cross product” ions are formed in the cases of the Cu metal-sodium butyrate and KClsodium butyrate mixtures. These results are direct evidence that ion-molecule reactions were not an important source of molecular ions in these cases. However, in the case of the CuClz-sodium butyrate mixture, there is significant formation of NaCuC4H702+ and the possibility of some formation of CuC4H702+. (Unfortunately, CuzO+ interferes with these peaks. However, CuzO+ is not observed in the copper powder mixture.) A possible conclusion follows that the CuClz and sodium butyrate react in the solid phase, evaporate as a molecular complex which during or after ionization undergoes decomposition to the observed NaCuC4H702+ ion. This interpretation is supported by the observation that significant amounts of NaCl+ are formed, which would require the postulation of either an ion-molecule reaction between sodium and chlorine or the same molecular complex postulated for the CuNaC4H702+. Since no peaks were observed a t m / e 78, 80 in the KC1 mixture, the occurrence of the required ionmolecule reaction seems unlikely. Received for review March 28, 1973. Accepted June 20, 1973. J. R. Hass acknowledges support by the Materials Research Center (University of North Carolina) under contract DAHC-15-67-C0223 with the Advanced Research Projects Agency. (38) A. J. Ahearn, Ed , “Mass Spectrometric Analysts of Solids,” Elsevier, New York, N Y., 1966, p 4
X-Ray Photoelectron Spectra of Lead Oxides K. S. Kim, T. J. O’Leary, and Nicholas Winograd Department of Chemistry, Purdue University, West Lafayette, Ind. 47906
ESCA spectra of seven different lead-oxygen species are examined including adsorbed oxygen on lead, two crystal forms of PbO, Pb304, PbO2, and two forms of adsorbed water on PbO2. The spectrum of Pb304 exhibits the behavior to two PbO molecules and one PbOz molecule although specific characterization of the crystal structure of PbO in Pb304 could not be made. The binding energies of the Pb 4f electrons in PbO2 are found to be lower than those of the Pb 4f electrons in PbO. The apparent reversal is rationalized in terms of a relaxation effect. Two kinds of adsorbed water are also found on Pb02. These are explained in terms of a surface adsorbed water
2214
species and a water or hydroxyl species which occupies lattice vacancies near the surface. These spectra were then used to characterize electrochemically generated PbOz. It is proposed that this technique may prove valuable in characterizing these electrochemically produced materials.
The technique of X-ray photoelectron spectroscopy (XPS or ESCA) has found extensive recent application in structural characterization of a variety of inorganic and organic compounds by appropriate correlation of measured electron binding energies of the various atoms in
A N A L Y T I C A L C H E M I S T R Y , VOL. 45, N O . 13, N O V E M B E R 1973
molecules ( I , 2 ) . Many operational complexities related to sample preparation, charging effects, and satellite lines have often tended to broaden the linewidth of measured peaks and alter peak positions. In addition, the escaping electron effectively carries information characteristic only of the first several molecular layers ( 3 ) . Samples whose surface structure is different from the bulk structure can therefore lead to erroneous conclusions about the bulk sample. A careful evaluation of the contribution of these effects to ESCA spectra is, of course, vitally important if the technique is to become a viable structural tool for bulk as well as for surface studies. We have been focusing our research on a detailed study of several metal-oxygen systems in an attempt to elucidate these factors (4-7).For example, the approach is well-suited to identifying the stoichiometry of oxides by comparison of binding energy chemical shifts to those of known compounds (4-6). The presence of chemisorbed gases on the sample can often lead to erroneous interpretation, particularly of 0 1s peaks, although recent efforts have successfully isolated the contribution of these species (5-7). Further studies such as the identification of crystal structures (6) and hydrates of surface and bulk species potentially offer another level of sophistication in the analysis of metal oxides heretofore unattainable. In this work we present the ESCA spectra for several different oxides of lead. By careful chemical preparation and characterization of various samples, s e w n different lead-oxygen species have been identified, illustrating the potential power that the ESCA technique may have in the analysis of complicated oxide films. Included in these species is the identification of two different crystal structures of PbO, chemisorbed oxygen atoms as well as two different forms of adsorbed water on PbOz. EXPERIMENTAL
1
, 1450'
Siegbahn et ai,, "ESCA: Atomic, Molecular, and Solid State Structure Studied by Means of Electron Spectroscopy," Almquist and Wiksells, Koktryckeri AB, Stockholm, 1967. (2) K. Siegbahn et a/., "ESCA Applied to Free Molecules," North-Holland, Amsterdam-London, 1969. (3) T. A. Carlson and G . E. McGuire. J . Electron Spectrosc., 1, 161 (1) K.
(1972/73). (4) K . S. Kim, N. Winograd, and R. E. Davis, J , Amer. Chem. Soc.. 93, 6296 (1971). (5) K. S. Kim and R. E. Davis, J. Electron Spectrosc., 1, 251 (1972: 73). (6) K. S. Kim and N. Winograd, Chem. Phys. Lett., 19, 209 (1973). (7) K. S. Kim, Phys. Rev.. submitted. (8) K. Siegbahn eta/., Science, 176, 245 (1972). (9) A. van der Drift. Philips Res. Rept. (Nefherlands), 21, 289 (1966). (10) M . R. Tomprett and J. St. Noble, Thin Solid Fiims, 5 , 81 (1970). (11) J. R. Anderson and V . 8. Tare, J . Phys. Chem., 66, 1482 (1964).
I
I370
-
'
5340
5300
5260
BINDING ENERGY(eV)
Figure 1. X-Ray photoelectron spectra of Pb 4 f 5 / 2 , 7 / 2 and 0 1s electrons of (a) orthorhombic PbO, ( b ) PbO2, and ( c ) Pb304 which were obtained commercially. Spectra of tetragonal PbO are similar to those of (a)
450
Apparatus. All spectra were recorded on a Hewlett-Packard 5950A ESCA spectrometer using monochromatic A1 Ka1.z X-rays obtained from a quartz-crystal disperser (8). The best possible full-width a t half-maximum (FWHM) recorded on the gold 4f7,2 level was 0.6 eV. The peak position of this line relative to the fermi level of gold was measured as 84.3 eV. Binding energies were, in general, reproducible to within *0.2 eV. The sample chamber was equipped with cross-probe allowing in situ deposition of metals onto the sample surface. The nominal pressure of this chamber was Torr while the spectrometer chamber normally had a residual pressure of 5 x 10-9 Torr as read on the ion pump gauge. Reagents. The PbO, Pb304, and PbOz were obtained commercially from Mallinckrodt and Baker and used without purification (Figure 1). Tetragonal PbO was prepared by evaporating PbO in cacuo on a hot (150 "C) plate (6, 9, I O ) and transferring the sample to the spectrometer in Ar or NZatmosphere (6) (Figure Pa). Rhombic PbO could be produced (6) by extensive exposure of the freshly evaporated lead surface to dry oxygen as confirmed by electron diffraction (11) or by exposure of the tetragonal PbO to
I
I410
41 0
370
5330
5290
5250
BINDIhG ENERGY(eV)
Figure 2. X-ray photoelectron spectra of Pb 4 f 5 / 2 , 7 / 2 and 0 1s electrons of (a) tetragonal PbO prepared by evaporation of PbO, ( b ) orthorhombic PbO prepared by exposing ( a ) to d r y oxygen, and ( c ) , sample ( b ) partially oxidized to PbO2, by exposing to oxygen ionized in an rf field (a) was observed to transform gradually to ( b ) ,and (c) was completely oxidized to PbOp after longer exposure to ionized oxygen oxygen (Figure 2 b ) . Many of these oxides could be produced by exposing samples to oxygen ionized by an rf discharge. Procedure. In attempting to evaluate binding energies for the various oxides of lead, charging of the nonconducting species always produced inconsistent results. These effects were compensated by employing several procedures. First, and most successfully, a thin discontinuous gold film was evaporated on the sample after the recommendatioiis of Hnatowich et al. ( 1 2 ) . Under these conditions, the gold film is isolated from the spectrometer by the sample. By referencing the observed lead 4f7,~and 0 Is peaks to the gold 4f712 peak, any charging in the sample would also appear in the binding energy of the gold peak. Since the gold 4f7,z electron binding energy was chosen to be 84.3 eV, any shifts in the lead or oxygen values can be appropriately corrected. This procedure could also be checked by evaporating lead on the lead oxide and making the same observations. In cases where the binding energy values obviously charged, the species could be identified by the difference in energy between the P b 4f7:2 peak and the 0 Is peak in the same sample (Table I ) . This difference was independent of all charging problems and often served as a convenient method of identifying certain species. In all cases reported in this work, the charging effect was less than 2 eV and produced no noticeable line broadening of spectra. ( 1 2 ) D. J. Hnatowich, J. Hudis, M . L. Appl. Phys., 42, 4883 (1971).
Perlman, and R. C. Ragaini. J.
A N A L Y T I C A L C H E M I S T R Y , V O L . 45, N O . 13, N O V E M B E R 1 9 7 3
2215
~
Table I. Binding Energies (eV) of Pb 4 f 7 / 2 and 0 1s Electrons for Various Lead-Oxygen Speciesa
Pb PbOads Pbo ( r h o m b i c ) PbO (tetragonal)
PbOz
136.9 137.6 137.9 137.9 ( P b 2 + ) 138.9 ( P b 4 + ) 137.9 137.5
... ...
aH2O bH2O
...
...
530.7 529.7 527.8
393.1 391.8 389.9 390.6 391.6 390.2 391.4 392.7
529'5 527.7 528.9 530.2
a Binding energies are calibrated to Au 4f7/2 electrons (84.3 eV) by evaporating Au on the samples (see text for details).
All spectra were deconvoluted u s i n g a D u P o n t 310 C u r v e R e solver. L o c a t i o n s a n d FWHM's for a species were f i r s t d e t e r m i n e d u s i n g spectra o f p u r e samples. L o c a t i o n s a n d FWHM's o f a species w h i c h was n o t o b t a i n e d as a p u r e species were a d j u s t e d until t h e b e s t fit was o b t a i n e d . S y m m e t r i c gaussian shapes were used in a l l cases.
RESULTS AND DISCUSSION Binding energies of P b 4f7/2 and 0 1s electrons for the various lead-oxygen species are given in Table I. Since the binding energies have all been calibrated to the evaporated gold 4f7/2 peak a t 84.3 eV, the given binding energies are believed to be free of the charging effect. The ESCA results for Pb, PbOad, (oxygen-chemisorbed lead), and rhombic and tetragonal PbO have been previously described elsewhere (6) and are included here for comparison. The spectra of PbO taken directly from the bottle (Figure l a ) yield identical curves for both crystal forms as indicated by the split 0 1s line. This result indicates the importance of characterizing the surface of the sample since, although visually the two forms are markedly different (the rhombic form is yellow and the tetragonal form is red), both crystal forms are present on the surface in approximately equal amounts on each pure bulk form. Yellow PbO freshly prepared by heating PbO2 in air a t 650 "C [the transition temperature (13) is 488 " C ] also shows spectra similar to those in Figure la. Only by careful preparation of the known species can these surface changes be characterized (6). The Pb 4f and 0 1s electron spectra of PbO2 are shown in Figure l b . Although the species is known to exist in two crystal modifications, a (rhombic) and (3 (tetragonal rutile), (14, we have not noted a measureable spectral difference between either form. For example, PbO exposed to an rf discharge of dry oxygen is expected to consist of both a and p forms of PbOz, (15), but only one 0 1s and P b 4f7,z line is observed for this sample. The fact that these forms are indistinguishable using ESCA is not surprising since in both forms each metal ion is in the center of a distorted octahedron of similar dimensions and since the difference in the crystal structures results mainly from (13) G. A. Bordovskii and V. A. Izvozchikov, Soviet Phys./Crystallogr., 12.818 11968). (14) See references cited in J. P. Carr and N. A. Hampson, Chem. Rev., 72, 679 (1972). (15) F. Lappe. J. Phys. Chem. Solid, 23, 1563 (1962). i
2216
~
~
I
~~~I
A N A L Y T I C A L C H E M I S T R Y , VOL. 45, NO.
different packing 'of oxygen octahedra (14). For the PbO case, however, substantial differences in bond distances and angles are noted between the rhombic and tetragonal structures (16) giving rise to distinct 0 1s peaks. Note that the P b 4f electrons of PbOz (Figure l b ) have a lower binding energy than those of PbO (137.5 us. 137.9 eV), a result not predicted by the normal correlation of binding energies to oxidation state of the atom. This binding energy reversal has been observed on PbO2 by Jorgensen (17) who reported 144.2 and 142.4 eV and by Melera (18) who reports 138.2 and 137.6 eV for PbO and PbOz, respectively. Similar observations of binding energy reversal have been reported for TI203 relative to T1 metal (19). This anomaly in measured binding energy is difficult to assess when comparing values for compounds of drastically different structure since the measured chemical shift of the electron binding energy is influenced by the Madelung lattice energy ( I , 20, 21) as well as by the electronegativity. However, since this binding energy reversal has been observed to date only in highly conductive oxides such as PbOz and Tho3, the effect of conductivity on atomic relaxation mechanisms may be significant. For example, if more effective relaxation can occur in the electron orbitals of conductive oxides, the binding energy reversal may be observed since photoionization is an adiabatic process and the relaxation energy is carried away by the departing electron (22, 23). The ESCA spectrum of Pb304 as shown in Figure I C shows P b 4f bands which can be deconvoluted into two peaks with an intensity ratio of 1:2 assuming symmetric gaussian peaks. The 0 1s line, however, is very broad and structureless. Assuming the lead species with the lower P b 4f binding energy is the Pb4+ ion, the spectrum is consistent with the stoichiometric structure (24, 25) PbO2.2PbO. Since in the lead-oxygen system ESCA could distinguish between different crystal structures of PbO ( 6 ) , we hoped to glean this sort of information about the environment of PbO in Pb304. As is shown in Table I, however, the interaction of PbO and PbO2 has clearly perturbed the observed binding energy from that of the pure molecules PbOz and PbO. For example, the difference in Pb 4f7,2 and 0 1s binding energies for Pb2+ in Pb&4 falls in between the two crystal structures of PbO, it is difficult to establish the exact structural form. This type of structural alteration has been observed using X-ray (24) and neutron (25) diffraction where attempts have been made to compare the free PbO2 and the free PbO (rhombic or tetragonal) to their structure in Pb304. Although these workers did not observe any major changes in bond lengths, various bond angle alterations led them to propose a different hybridization for each form. This observation did not lead, however, to a consistent conclusion concerning the crystal structure of PbO in Pb304 and the details concerning the interaction remain open. (16) M. I. Kay, Acfa Crystallogr., 14, 80 (1961). (17) C. K. Jorgensen, Theor. Chim. Acta.. 24, 241 (1972). (18) A . Melera, Hewlett-Packard Corporation, private communication, 1972. (19) J. Current, Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, Cleveland, Ohio, 1972, Paper No. 236. (20) C. S. Fadley, S. 8. M. Hagstrom, M. P. Klein, and D . A. Shirley, J. Chem. Phys., 48, 3779 (1968). (21) R . R . Slater, Surface Sci.. 23, 403 (1968). (22) H. W. Meldner and J. D. Perez, Phys. Rev.,PartA, 4, 1388 (1971). (23) D. W . Davisand D. A. Shirley, Chem. Phys. Lett., 15, 185 (1972). (24) B. Dickens,J. lnorg. Nucl. Chem., 27, 1509 (1965). (25) M. K. Fayek and J. Leciejewicz, Z. Anorg. Allg. Chem.. 336, 104 (1965).
13, NOVEMBER 1973
With the ESCA spectra of the various lead oxides wellcharacterized, we have tested the technique to identify unknown products formed during the vigorous oxidation of various lead species. For example, a rhombic PbO surface could be oxidized by exposure to oxygen ionized in an rf Torr, field (Figure 2c). In a pure oxygen discharge at a thin film of PbO2 was produced, but when the oxygen pressure was reduced to about 10-3 Torr, the surface was reduced back to polymorphic PbO. No evidence for the presence of intermediate oxides like Pb304 was found and only high oxygen partial pressures were successful in generating the PbO2 surface films. The PbsOC also could be oxidized to PbO2 by the same method. Although electron diffraction (15) and X-ray diffraction (26) yield similar information on much thicker surface layers, the higher sensitivity of ESCA is clearly illustrated in this case. By comparing the relative peak areas of the P b 4f7/2 electrons in PbO and PbO2 (Figure 2c) and considering the mean escape depth of 1200-eV electrons to be about 15 A ( 3 ) ,we estimate the surface is covered with the equivalent of one monolayer of PbO2. Examination of the 0 Is spectrum of PbO2 also allows investigation of various hydration processes which occur near the sample surface. For example, the rf-produced P b 0 2 was treated with a drop of water yielding two additional 0 1s peaks a t 530.2 and 528.9 eV (Figure 3b). If this film is stored in a water saturated atmosphere for several days, the 0 Is peak with the lower binding energy (528.9 eV) predominates (Figure 3c). These 0 Is peaks can be regenerated by further treatment with the rf discharge followed by another exposure to water. The commercial samples of PbO2, however, do not yield the intense 0 1s peaks shown in Figure 3b and c. Even the spectra of these samples which we soaked in water for several days produce a signal identical to the one shown in Figure l b . The interaction of water with PbOz has been related to the formation of oxygen vacancies in Pb02. Mindt (27) suggests that the generation of oxygen vacancies (1) (202-)lattice + ( 2 0 0 + 4e-)lattice + ( 0 2 ) g a s may be assisted by surface adsorbed water, and according to Duisman and Giauque (28), water molecules may be firmly bound by occupying oxygen vacancies in the lattice. An additional model based on the replacement of oxygen ions with hydroxyl ions due to the oxidation of adsorbed water, i. e , (402-)lattlce+ (2H2O)ads (40H-
-+
4e-)lattice +
(0,)gas
(2)
has also been proposed (28), although the details of the effect remain in doubt. Although the ESCA results cannot distinguish at this time whether reaction 1 or reaction 2 is the predominant pathway, several conclusions can be made concerning the interaction of water with the PbO2 surface. For example, microanalysis of the sample in Figure 3b and c yields a total hydrogen concentration of less than 0.02% and mass spectrometric analysis at 250 "C shows no significant increase of OH2f above background. The ESCA data in Figure 3b show that the additional oxygen containing species and PbO2 are present in nearly equal amounts.
(26) A. Czapla. M . Jachimowski, and E. Kusior, Acta. Physioi. Poi., Part A , 41, 149 (1972). (27) W. M i n d t . J . Electrochem. SOC., 116, 1076 (1969). (28) J . A. Duisman and W. F. Giauque. J. Phys. Chern., 72, 562 (1968).
I
1
5340
5300
I
1
5260
BINDING ENERGY(eV)
Figure 3. X-Ray photoelectron spectra of 0 1s electrons of ( a ) Pb02 prepared by exposing PbO powder (Figure l a ) to oxygen ionized in an rf field, ( b ) sample ( a ) treated with a drop of water, (c) sample ( b ) kept in a H2O-saturated atmosphere for 3 days, and ( d ) PbO2 prepared electrochemically by oxidation of P b ( N 0 3 ) 2 on a Pt electrode at 1.5 V vs. SCE. 0 1s peaks due to adsorbed water in (c) disappeared after an rf treatment. The Pb 4f lines are unaffected after the adsorption of water
Since ESCA is sensitive only to the first few angstroms of the sample, we must conclude that both oxygen species not attributed to PbO2 are present at or near the surface. Further, since the species with an 0 Is peak at 530.2 eV is not a stable form ( i e . , it converts to the 528.9 eV species in several hours) and since this form always appears to be a prerequisite form for the formation of the 528.9 eV species, we ascribe it to adsorbed water interacting with available surface oxygen vacancies. This assignment is consistent with either reaction 1 or 2 if the 528.9eV species is considered to be hydroxyl or water incorporated into the lattice near but not at the sample surface. The variation of activity to adsorbed water on the commercial samples is difficult to assess due to the complicated microscopic properties of an ill-defined surface. The generation of active sites, presumably surface oxygen vacancies, by high energy particles like O2f present in the rf discharge provides a logical explanation for the increased activity to water adsorption for the rf-produced PbO2. This idea is further supported by the fact that the 530.2-eV form can only be regenerated by a new exposure to 0 2 + followed by treatment with water. The 0 Is spectrum of PbOz produced by the electrochemical oxidation of P b ( N 0 3 ) ~is given in Figure 3d. Clearly a major contribution to the observed peak is due to water incorporated into the lattice of the resulting PbO2 film. The distribution of the water again must be near the surface since, according to microanalysis of a similarly produced sample, the hydrogen concentration is less than 0.1% (29). Further studies aimed at characterizing PbOz films produced electrochemically under a variety of experimental conditions are in progress.
(29) N. E. Bagshaw, R. L. Clarke, and B. Halliwell. J , Appi. Chern.. 16, 180 (1966).
ANALYTICAL C H E M I S T R Y , VOL. 45, NO. 13, N O V E M B E R 1973
2217
CONCLUSIONS
ACKNOWLEDGMENT
The above study on the various lead-oxygen species points out the power the ESCA technique possesses in the analysis of complicated surface films. These studies also illustrate a number of pitfalls regarding the use of ESCA as a structural tool for analysis of bulk samples. The presence of different crystal structures may produce anomalous peaks in spectra. Adsorbed gases or water on the surface can broaden peaks and mask important structure in the observed peaks. The surface sensitivity of ESCA can certainly be advantageous in studying these adsorbed materials although they can often complicate the analysis of bulk sample characteristics.
The assistance of Jon Amy and William Baitinger is acknowledged.
Received for review April 4, 1973. Accepted June 4, 1973. The authors wish to thank the National Science Foundation (Grant No. GP-37017X) and the Air Force Office of Scientific Research (Grant No. AFOSR-72-2238) for financial support. The authors also appreciate the equipment grant from the National Science Foundation which allowed purchase of the ESCA instrument.
Equipment Stability in X-Ray Fluorescence Spectrometry and Radioactive Counting-A Statistical Approach K. A. H. Hooton and M. L. Parsons' Department of Chemistry, Arizona State University, Tempe, Ariz. 85287
Unstable conditions in the component units of X-ray fluorescence spectrometers and radioactivity counting equipment can result in serious errors of analysis. The mean square successive difference (MSSD) method is introduced for the first time to reveal drift or oscillations. Drift and large oscillations present together can be detected by new statistics developed from the MSSD method. The chi-squared test, analogous to its use in radioactivity counting, has been adopted for X-ray fluorescence spectrometer stability testing and is used in conjunction with the MSSD method. These combined methods are more sensitive than the present criterion for acceptable equipment stability in which the set of observed replicate counts is expected to lie within the range of f3 u about the mean. Drift often can be detected with fewer observations than previously and abnormal distributions of observations, not found by the 3 u criterion, are detected. The combined statistical tests are applied expeditiously by means of a desk calculator.
the same manner as a radioactive source. Liebhafsky and coworkers (2) experimentally showed that the frequency of X-ray counts from their X-ray fluorescence spectrometer and the frequency of alpha particle counts obtained by Rutherford and Geiger (3) both fit the same Gaussian curve. Because of its greater complexity, the X-ray fluorescence spectrometer has a greater risk of counting errors or instability than the radioactivity counter. For an infinite number of observations (total counts, not count-rate) in equal counting periods, two important parameters are the population variance U Z and the population standard deviation u. The best estimate of u2 is the sample variance s2 = Z(X,
- X)Z/(n - l),
1
(1)
=
1,2, ... n
(1)
where x L = the value of the ith observation, and f = mean of the n observations. Established theory also shows ( 1 ) that, for the distributions and conditions which are of concern here, the population variance g2
The radioactive decay of atoms is a random process and the number of observed disintegrations occurring in equal short intervals of time has a binomial frequency distribution ( I ) . The Poisson and normal (or Gaussian) frequency distributions closely approximate the binomial distribution ( I ) under certain conditions which generally are easily met in radioactivity counting (Symbols and definitions of statistical terms appear in the Appendix.). The counts obtained from the X-ray fluorescence spectrometer during a number of fixed time intervals also will closely approximate the same distribution so that the sample behaves in
i
= p
( 2)
where = the mean of the population. Generally, u2 is estimated by (T?
=
-
x
(3)
where f , the sample mean, or mean of the series of observed counts, is the best estimate of F . It must be emphasized that the value of s2 is correctly evaluated by Equation 1 whether or not the equipment is stable, whereas Equation 3 is correct only for stable equipment. For a normal distribution. 68.3% of the observations should fall within *1 standard deviation (std dev) of the mean. 95.5% within zk2 std dev and 99.7% within zk3 std
A u t h o r t o w h o m a l l correspondence s h o u l d b e addressed.
G. Friedlander, J. W. Kennedy, and J. M. Miller, "Nuclear and Radiochemistry," 2nd ed., John Wiiey and Sons, New York, N . Y . , 1964, Chap. 6.
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NO.
(2) H. A. Liebhafsky, H. G. Pfeiffer. and P. D. Zemany. Anal. Chem., 27. 1257 (195.5). (3) E. Rutherford and H. Geiger, Phil. Mag., 20, 698 (1910).
13, N O V E M B E R 1 9 7 3
