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- https://ufdc.ufl.edu/UF00097950/00001
## Material Information- Title:
- The calculation of limits of detectability and optimum conditions for atomic emission and absorption flame spectrometry
- Creator:
- Vickers, Thomas J., 1939-
- Place of Publication:
- Gainesville
- Publisher:
- [s.n.]
- Publication Date:
- 1964
- Copyright Date:
- 1964
- Language:
- English
- Physical Description:
- xii, 124 l. : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Atoms ( jstor )
Combustion temperature ( jstor ) Flames ( jstor ) Line spectra ( jstor ) Mathematical optima ( jstor ) Monochromators ( jstor ) Signal detection ( jstor ) Signals ( jstor ) Thermal radiation ( jstor ) Wavelengths ( jstor ) Absorption spectra ( lcsh ) Chemistry thesis Ph. D Dissertations, Academic -- Chemistry -- UF Flame photometry ( lcsh ) Radiation ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis - University of Florida.
- Bibliography:
- Bibliography: l. 121-123.
- Additional Physical Form:
- Also available on World Wide Web
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000423912 ( AlephBibNum )
11026624 ( OCLC ) ACH2317 ( NOTIS )
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THE CALCULATION OF LIMITS OF DETECTABILITY AND OPTIMUM CONDITIONS FOR ATOMIC EMISSION AND ABSORPTION FLAME SPECTROMETRY By THOMAS JOSEPH VICKERS A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA April, 1964 ACKNOWLEDGMENTS It is with sincere pleasure that the author takes this opportunity to acknowledge his debt of gratitude to his research director, Dr. J. D. Winefordner. The author is indebted to Dr. Winefordner for his instruc- tion, encouragement, and advice, and many hours of his time spent in fruitful discussion of the material presented in this dissertation. The author also wishes to express his gratitude to Rev. William J. Rimes, S. J., of Spring Hill College. It was his assistance and dedicated teaching which encouraged the author to pursue higher studies in chemistry. TABLE OF CONTENTS Page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES KEY TO SYMBOLS Section I. INTRODUCTION II. CALCULATION OF THE LIMIT OF DETECTABILITY FOR ATOMIC EMISSION FLAME SPECTROMETRY Introduction Derivation of Equations Discussion and calculations III. CALCULATION OF THE LIMIT bF DETECTABILITY FOR ATOMIC ABSORPTION FLAME SPECTROMETRY Introduction Derivation of Equations Discussion and Calculations IV. CALCULATION OF OPTIMUM CONDITIONS Introduction Selection of Optimum Slit Width Effect of Flame Conditions on Atomic Concentration Optimum Flame Conditions for Atomic Emission Flame Spectrometry Optimum Flame Conditions for Atomic Absorption Flame Spectrometry Calculations Experimental Verification of Theory V. CONCLUSIONS APPENDICES LITERATURE CITED BIOGRAPHICAL SKETCH 124 LIST OF TABLES Table Page 1 REPRESENTATIVE RESULTS IN ATOMIC EMISSION FLAME SPECTROMETRY FOR LIMITS OF DETECTABILITY IN NUMBER OF ATOMS PER CM.3 OF FLAME GASES FOR TWO ELEMENTS (Na and Cd) IN TWO FLAMES AND FOR TWO MONOCHROMATORS WITH OPTIMUM SLIT WIDTHS 17 2 CALCULATED VALUES OF OPTIMUM SLIT WIDTHS IN CM. AND LIMIT OF DETECTABILITY IN ATOMS PER CM.3 FOR SEVERAL ELEMENTS WITH RESONANCE LINES IN WIDELY DIFFERENT SPECTRAL REGIONS AS A FUNCTION OF FLAME TEMPERATURE FOR A REPRESENTATIVE MONO- CHROMATOR-DETECTOR SETUP 21 3 VARIATION OF Nm WITH W FOR A PRISM MONOCHROMATOR AT 420 mp WHEN USING AN OXYHYDROGEN FLAME 24 4 REPRESENTATIVE RESULTS IN ATOMIC ABSORPTION FLAME SPECTROMETRY FOR LIMITS OF DETECTABILITY IN ATOMS PER CM.3 OF FLAME GASES FOR Na AND Cd IN TWO FLAMES 36 5 VARIATION OF Nm WITH W FOR SEVERAL MONOCHROMATORS AND FOR SEVERAL FLAME TYPES 39 6 OPTIMUM SLIT WIDTH AND TOTAL ROOT-MEAN-SQUARE NOISE FOR THREE FLAMES 62 7 VALUES OF Kl, K2, AND K3 AS A FUNCTION OF T 74 8 PER CENT ERROR IN N AND ABSORBANCE VALUES DUE TO FINITE WIDTH OF SOURCE EMISSION LINE AS COMPARED TO WIDTH OF ABSORPTION LINE 96 9 HYPERFINE STRUCTURE COMPONENTS OF THE Na 5890 A. LINE 101 LIST OF FIGURES Figure Page 1 Calculated Plots of Atomic Concentration of Sodium Versus Flame Temperature for Several Total Pressures of Sodium and for Several Flame Types. 54 2 Calculated Plots of Signal-to-Noise Ratios for the Na 5890, 5896 A. Doublet Versus Flame Temperature for Several Total Pressures of Sodium and for Several Flame Types. 64 3 Calculated Plots of NL/T1/2 Versus Flame Temperature for Several Total Pressures of Sodium and for Several Flame Types. 70 4 Calculated and Experimental Analytical Curves for the Na 5890, 5896 A. Doublet for Several Flame Types. 76 5 Calculated and Experimental Plots of Signal-to- Noise Ratio Versus Slit Width for Sodium in a Stoichiometric H2/02 Flame. 82 KEY TO SYMBOLS A effective aperture of the monochromator, cm.2 a damping constant, no units A absorbance, no units AT total absorption of spectral line, no units At transition probability, sec.-1 Aa 1 I/I1, no units A9 atomic absorptivity at frequency V, no units B factor characteristic of photodetector surface, no units Be rotational constant of molecule cm.-1 B(T) partition function, no units C concentration, moles/liter c speed of light, 3 x 1010 cm./sec. Ci constant defined in text D angular dispersion, radians/mp DNaCl dissociation energy of NaC1, electron volts e base of natural logarithms, no units Ei energy of state i, electron-volts ec electronic charge, 1.59 x 10-19 coulombs ef expansion factor, no units F focal length of collimator, cm. Af frequency response band width, sec.- G gain per stage of photomultiplier, no units gi statistical weight of state i, no units gol statistical weight of ground state of ion, no units go* statistical weight of ground state of molecule, no units H slit height, cm. h Planck's constant, 6.62 x 10-27 erg-sec./atom I integrated intensity, watts/cm.2 ster. i output signal, amperes Io transmitted intensity with no sample in flame, watts/cm.2 ster. iO signal due to IO, amperes AIo r.m.s. fluctuation in Io, watts/cm.2 ster. sec.-1/2 iTo noise signal due to fluctuations in I1, amperes 2 Ic intensity of the flame continuum, watts/cm.2 ster. mp ic signal due to Ic, amperes ZIc root-mean-square fluctuation of intensity of flame continuum, watts/cm.2 ster. mp sec.-1/2 Zic noise signal due to fluctuation of flame continuum, amps. id dark current, amps. Ie intensity of thermal emission of the line of interest, watts/cm.2 ster. ie signal due to le, amperes 2 -1/2 AIe r.m.s. fluctuation in Ie, watts/cm.2 ster. sec.- Aie noise signal due to fluctuation in Ie, amperes If intensity of fluorescent emission of the lines of interest, watts/cm.2 ster. if signal due to If, amperes ii chemical constant of species i, no units Im intensity due to N watts/cm.2 ster. im output signal due to Nm, amps. Zip phototube noise signal, amps. vii is signal due to scattered incident radiation, amperes AIS r.m.s. fluctuation in intensity of scattered radiation, watts/cm.2 ster. sec.-1/2 Lis noise signal due to fluctuation in scattered incident radiation, amperes It transmitted intensity with sample in flame, watts/cm.2 ster. it signal due to It, amps. AiT total noise signal, amps. I(B intensity of a black body radiator at frequency V watts/cm.2 ster. B I o intensity of a black body radiator at frequency Vo, watts/cm, ster. k Boltzmann constant, see units in text kO atomic absorption coefficient at the line center, cm.-1 k0m atomic absorption coefficient at the line center at the minimum detectable concentration, cm.-1 ko* atomic absorption coefficient at the line center for a pure Doppler broadened line, cm.-1 Kc constant defined in text KH dissociation constant for HC1, atm. Ks constant defined in text kc k8 kg/AiT kd k8 kll/AiT K1 dissociation constant for NaC1, atm. K2 ionization constant for Na, atm. K3 dissociation constant for NaOH, atm. kI CTfH(A/F2) nI k2 2BMecAf k3 YTfH(A/F2) Ic k4 YTfH(A/F2)ic k5 TfH(A/F2) Io (1 e-koL) viii kg Tf H(A/F2) k7 (2BMecI)/( T HA/F2) 7 2B^cid 1 k8 ITfH(A/F n (TH(A/F2) (Af)l/2 R)2 h Vogu k9 47 x 107B(T) At hV 2 VD gu 1/2 ( At a) 10 107 c g rnn2 B(T) h A o At 11 lo 0 4B(T) (11 22) h 2 2 AVD a At 1/2 /2] k12 107 c ,-ln2 B(T) L 1 (gl) 2(g2) k13 [( oI/IO)(Af)1/2 1 1 ___ 1/2 k ,2 2 X o2guAt c Ma 14 -R 8 x B(T) 2 12 R in 2 o k 82 in 2 Xo2(g+g2)c Mal/2 AtJ 15 ,I 8j( B(T)2 42 R in 2 o' k atomic absorption coefficient at9 other than ),, cm.-i L flame diameter, cm. M amplification factor of photodetector, no units Ma atomic weight, atomic mass units Mf effective molecular weight of foreign species, atomic mass units MNaCl N n N Nf Ni molecular weight of NaC1, atomic mass units total atomic concentration of species of interest, atom/cm.3 entrance optics factor, no units ground state concentration of species of interest, atoms/cm.3 concentration of foreign species, particles/cm.3 atomic concentration at intersection (self-absorption) point, atoms/cm.3 Nm minimum detectable number of atoms/cm.3 of flame gases Nmo nTI "298 P Pa PC Pe Pe Pf Pi PT Q R Rd Rf Ro s sa sc Sd Sm T AT Tf Tm m To V ground state minimum detectable concentration, atoms/cm.3 moles present at temperature T and temperature 2980K., respectively power, watts pressure of species of interest, mm. partial pressure of C1 in all forms, atm. partial pressure of electrons in the flame, atm. partial pressure of electrons due to ionization of flame gases, atm. pressure of foreign species, mm. partial pressure of species i, atm. total pressure of species of interest in all forms, atm. flow rate of unburned gases, cm. /sec. gas constant, 8.3 x 107 ergs/moleoK. reciprocal linear dispersion, mp/cm. reflectance of optics, no units resistance of load resistor, ohms spectral slit width, mi spectral slit width due to aberrations, coma, etc., mp sd + sa' mP diffraction limited spectral slit width, mp spectral slit width as determined by mechanical slit width, mp flame temperature, OK. root-mean-square temperature fluctuation, OK. transmission factor of optics, no units transmittance of flame, no units temperature of load resistor ionization energy of atom, electron volts v 2( Vo) in no units A D W slit width, cm. Wo optimum slit width, cm. y 2 i ln 2 no units AVD ZH number of Holtsmark broadening collisions/sec. atom ZL number of Lorentz broadening collisions/sec. atom a ratio of emission line width to absorption line width, no units co width of beam of radiation at dispersing element, cm. P fraction to account for incomplete compound formation and atomic losses due to ionization, no units 'i abundance of isotope i, no units Sphotosensitivity factor, amps./watt A a variable distance from the point 9 Vo, sec.- d factor to account for line broadening other than Doppler broadening at 9 = ) o, no units SV factor to account for line broadening other than Doppler broadening at 9 other than V) no units 6 atomization efficiency, no units a statistical weight fraction, no units S ratio of intensity of hyperfine component i to the sum of the intensities of all other hyperfine components due to nuclear spin, no units correction factor to account for multiplicity of spectral line, no units slit function parameter, no units X wavelength setting of monochromator, cm. Xo wavelength at line center, cm. Q frequency, sec.-1 Qo frequency at line center, sec.- o frequency at line center, sec. V ratio of root-mean-square fluctuation in the background intensity to background intensity, sec.1/2 t 3.14, no units /0 correction factor to account for hyperfine structure of source and absorption line, no units cross section for Holtsmark broadening, cm.2 0l cross section for Lorentz broadening, cm.2 T lifetime of an excited state, seconds 0 flow rate of solution, cm.3/min. 1/2 X AIb/I, sec. P --e/Ie, sec.1/2 We vibrational constant of molecule, cm.- wi TiLk, no units I. INTRODUCTION The selection of optimum conditions is a problem which confronts every analyst. To the analyst using atomic emission and atomic absorp- tion flame spectrometry this problem is particularly vexing. Trial- and-error methods of choosing optimum conditions are tedious and often misleading because of the large number of variables and their inter- dependence, and, until the present time, no adequate treatment has been given describing the quantitative relationship of experimental factors to the sensitivity of analysis. The work reported here was begun as an attempt to take a more systematic and quantitative approach to the question of optimum conditions for flame spectrometric analysis. How- ever, the theoretical principles and expressions developed in this paper have wider applicability than the prediction of optimum conditions and, in fact, should provide the means for a quantitative discussion of many of the phenomena and interference so frequently encountered in flame spectrometry. It is hoped that two major purposes will be served by this study: first, to demonstrate that the influence of experimental variables on the measured signal in atomic emission and atomic absorption flame spectrometry can be treated quantitatively in a relatively simple manner, and, second, that from this treatment optimum conditions of analysis can be selected. It should be clear that if these purposes are achieved this study will provide information of aid in routine analysis as well as in theoretical considerations. The approach taken in this study is to write expressions for the signal and the noise as functions of the experimental parameters for atomic emission and atomic absorption flame spectroscopy. The limit of detectability is then defined as the atomic concentration of the species of interest which gives a signal-to-noise ratio equal to two. The resulting expression for the limit of detectability makes possible the estimation of the effect of various experimental parameters on the sensi- tivity of analysis and also allows a direct calculation of the limit of detectability which may be expected. The selection of optimum condi- tions for any concentration range of the species of interest is dis- cussed on the basis of a maximum signal-to-noise ratio. The effects of compound formation and ionization of the species of interest and self- absorption of radiation are considered. Using the derived expressions the calculation of optimum conditions is carried out for the atomic emission flame spectrometric analysis of sodium, and the results of the theory are compared with experimental measurements. II. CALCULATION OF THE LIMIT OF DETECTABILITY FOR ATOMIC EMISSION FLAME SPECTROMETRY Introduction The system under consideration in the development of this theory consists of a flame source, a lens or mirror for focusing the emitted radiation on the monochromator entrance slit, a monochromator with accompanying optics and slits, a photodetector, amplifier, and readout (recorder, meter, etc.). For the purposes of the theory, it is assumed that the source is aligned for maximum intensity and that the slit is fully and uniformly illuminated. The following discussion will apply to either flames produced using total consumption or chamber type atomizer- burners. However, when using atomizer-burners, flames are far from homogeneous, and so it will be assumed that the brightest part of the outer cone of the flame is always selected for viewing. For purposes of simplicity the dimensions of the entrance and exit slits are assumed to be the same, although, where the two slits are separately adjustable, a small gain in sensitivity may result by having the exit slit slightly larger than the entrance slit. The width and height of the slits are, respectively, W and H. The effective aperture of the monochromator is 2 A, cm. and the focal length of the collimator lens or mirror is F, cm. The term Tf is the transmission factor of the spectrometric system (includes entrance optics as well as monochromator optics) as determinedby absorption and reflection losses at all optical surfaces. The spectral slit width of the monochromator is denoted s and is given (6,8,29) approximately by s = sm + Sd + sa sm + Sc [11 where sm is the spectral slit width of the monochromator when using wide slits, sd is the diffraction limited spectral slit width of the mono- chromator assuming focusing aberrations are negligible, sa is the spectral slit width of the monochromator due to coma, aberrations, imperfect optics, mismatch of slit curvature, etc., which results in a circle of.confusion of constant size at the exit slit, and sc sd + sa. In this paper the spectral slit width will always have units of mp. The terms sm in mp is given (8,29) by sm = (1/DF)W = RdW, [2] where D is the angular dispersion of the monochromator in radians per mp, and Rd is the reciprocal linear dispersion of the monochromator in mp per cm. (i.e., A. per mm.). The term sd, in mp, is given (8,29) by sd (1/D)X1/c FRdX/o [3] where X is the wavelength setting of the monochromator in cm., and 06 is the width of the beam of radiation in cm. at the dispersing element. The term sa, in mp units, is a constant characteristic of the mono- chromator. Equation [1] is not strictly correct because the above effects are not strictly additive (6). However, equation [1] will give the correct value of the spectral slit width when any one of the terms predominates. If the above effects are of the same magnitude and are independent, then the spectral slit width, s, will be given more accurately by Pythagorean addition, i.e., s2 = Sm2 + Sc2. In this section the additive relationship for s, given in equation [1], will be used. In addition to the above requirements concerning the entrance optics and the monochromator, the following requirements will be made for the system under consideration. It will be assumed that the detector intercepts all the radiation passing through the exit slit and that the amplifier-readout system is well regulated so that the limiting noise is a result of detector and flame noise. The above assumptions and require- ments regarding the instrumental system to be used are generally valid for any good commercial flame spectrometer; and, therefore, the following discussion should be directly applicable to many experimental systems. Derivation of Equations The minimum detectable concentration, N will be defined as that concentration which produces an average output current anodicc current of phototube) im, in amperes, such that' im = 2 A [4] where AiT is the root-mean-square fluctuation in the background anodic current of the phototube. The value of i is given (20) by im = YTfLmWH(A/F2)n [51 where y is the photosensitivity factor, i.e., the current in amperes produced at the anode of the detector for each watt of radiant power incident upon the photocathode, and Im is the total intensity (integrated intensity) of the spectral line (in watts/cm.2 ster.) produced by the minimum detectable concentration. (More correctly, Im should be called the steradiancy (32) of the spectral line, but the more general term "intensity" is used throughout with the units carefully specified.) The term A/F2 is the number of steradians viewed by the spectro- metric system as long as the effective aperture is filled with radiation. This is true whether the source is viewed directly or a lens or mirror is used to focus a selected portion of the flame on the slit. The para- meter n is the number of solid angles of value A/F2 which are gathered into a single solid angle by means of a suitable arrangement of entrance optics. As has been pointed out by Gilbert (17), it is possible by the use of a suitable system of mirrors to increase the intensity incident upon the spectrometric system. Thus, if a mirror is suitable placed so as to focus an additional image of the source on the slit, then the total incident intensity is increased by n = 1 + RfTm where Rf is the reflec- tance of the mirror and T is the transmittance of the flame for the m particular spectral line in concern. Using the spectrometric system which is described at the beginning of this section, a single viewing (n 1) of the flame is assumed. Once a theory is worked out for a single viewing of the flame source, it is easy to evaluate n for any number of viewings by proper consideration of the geometry of the entrance optics. Gilbert (17) has considered several possible optical arrangements for increasing the light gathering power. The parameter V accounts for the position of the spectral line with respect to the slit function distribution curve. For equal entrance and exit slits, the distribution curve is triangular, and the slit function parameter is given (8) by 1 [61 s where X is the wavelength setting of the monochromator, and ko is the wavelength of the line center. For monochromator wavelengths greater than Xo+ s and less than Xo s, fCis zero. If the monochromator has sufficient resolution to isolate a single, sharp spectral line, then the value of ( can be made equal to unity by adjusting the monochromator wave- length, X, to the peak wavelength of the spectral line, X0. The integrated intensity Im in units of watts/cm.2 ster. is given (2) by the well known equation Im (10-7/4[)NmLh]oAt[gu/B(T)]eEu/kT [7 where Nm is the minimum detectable number of atoms/cm.3 of flame gases, h is Planck's constant in ergs/sec., the 10-7 is necessary to convert from ergs/sec. to watts, 0o is the frequency of maximum intensity of the spectral line in sec.-1, At is the transition probability in sec.-l for the transition from the upper state u to a lower state (usually the ground state of the atom, designated o), Eu is the energy of the upper state, k is the Boltzmann constant in units consistent with Eu, T is the absolute temperature, and L is the average diameter in cm. of the flame region being focused on the monochromator entrance slit. The statistical weight of the upper state is gu, and the partition function over all states is B(T), which is defined by B(T) go + gleE/kT + g2-E2/kT.... e-Ei/kT [8 where the summation is carried out over all states. However, only in cases in which there are excited states within 1.5 to 2 e.v. of the ground state (e.g., Cr at 30000K in which B(T) is about 9 per cent greater than go) does B(T) differ significantly from go and so in most cases the simplificationB(T) = go can be made with negligible error. More correctly, Nm in equation [7] should be replaced by Nm, the ground state atomic concentration at the limit of detectability. The minimum o 0 detectable concentration, Nm, is related to Nm by the expression N = Nmgo/B(T), where, as discussed above, B(T)Sgo for most cases in flame spectrometry. As long as the spectral line is single, sharp, and isolated, the value of Im is given by equation [7]. If radiation from more 'than one line of the same element is passed by the spectroscopic system, then Im- must contain additional terms for each line similar to the right hand term of equation [3]. In this case im will be given by an equation similar to equation [5], namely, im TfWH(A/F2)n `djIjj [9] where the summation is over all spectral components passed by the exit slit. Since all components will not be passed centrally through the exit slit, qj, the slit function for each spectral component passed by the exit slit, will be less than unity for each line. The photosensi- tivity factor, jj, for each spectral line passed by the exit slit will, however, be nearly the same as the photosensitivity factor for the wavelength at which the monochromator is set as long as the change in ' with wavelength is small over the spectral slit width of the instrument. In any event it would be a relatively simple matter to determine the value of (j for each spectral line passed by the exit slit by means of ~_~ the manufacturer's spectral response curves for the photomultiplier tubes being used. The evaluation of fj for each spectral component can be performed by equation [6]. The greatest problem in evaluating the summation in equation [9] is in determination of the number of spectral components passed by the exit slit, i.e., as the slit width becomes wider more spectral lines will be passed, especially for such elements as the transition metals and the rare earth metals which have complex spectra. A special problem results if spectral lines have wavelengths near the extremes of the slit distribution. If the spectral line is so broad that the intensity read by the instrument is not the total line intensity, then additional factors must be included in equation [9]. This problem has been considered by Winefordner (41) and Brodersen (9). However, most spectral lines pro- duced by excitation of atoms in flames are quite narrow compared with the spectral slit width of most monochromators, and so additional correction factors are rarely required. It is not possible to consider in a general way the case in which the spectral line is not resolved from lines of other elements because this would require a detailed knowledge of the matrix in which the analysis is to be performed. In the following discussion, the spectral line will be considered single, sharp, and isolated, and the mono- chromator wavelength will be assumed to be adjusted to the line center (i.e., K,- 1). In this way the optimum slit width can be determined by minimization of the limit of detection equation, Im will be given by equation [7], and im will be given by 'im TfImWHA/F2 (t- 1 and n = 1). As noted in the above discussion, there are cases in which the assumption of a single, sharp, and isolated line is not valid. However, it should be pointed out that even if in a particular case a line does not meet the criteria of single, sharp, and isolated, it is still possible to obtain exact results by substitution of the proper values into the summation of equation [9]. More energetic sourcessuch as arcs, sparks, etc., certainly do not meet the above criteria. The root-mean-square fluctuation current, AiT, is a result of the root-mean-square fluctuation current due to noise in the flame source, Aic, and the root-mean-square fluctuation current due to noise in the phototube, Aip. The flame noise is primarily due to fluctuation in the intensity of the continuous background radiation, and the photodetector noise is due to the shot effect and thermal noise. The two noise signals add quadratically (19) so that AiT is given by Ai (Ap2 + rc2)1/2 [10] The value of Aip is given (35) by 1/2 [ ip (2ecAfBM(id + ic) + 4kTof/Ro) [ where ec is the electronic charge in coulombs, B is a constant approxi- mately equal to 1 + 1/G + 1/G2 + 1/G3 + .... where G is the gain per stage of the photomultiplier tube, M is the total amplification factor -1 of the phototube, and Af is the frequency response bandwidth in sec.- of the amplifier-readout circuit. Both flame noise and phototube noise are assumed to be equally distributed over Af, i.e., the noise is assumed to be white (19). The dark current in amperes at the anode, produced by thermionic emission, is id, and the signal in amperes pro- duced at the anode due to the incident intensity of continuous radiation from the flame is ic. The temperature of the load resistor in OK is To, and the load resistor of the detector circuit has a resistance of Ro ohms. The value of ic in amperes is given (see Appendix A) by ic yTfIcWH(A/F2)s [12] where Ic is the intensity per unit wavelength interval of the continuum, i.e., Ic has units of watts/cm.2 ster. mp (called steradiancy per unit wavelength interval). The value of Aic can also be shown to be given (see Appendix A) by Aic ~TfAIcWH(A/F2)s(f)12 [ 13] where AIc is the root-mean-square fluctuation in the intensity (or steradiancy) per unit wavelength interval of the continuum. Combining several of the previous equations and solving for Nm, one may write 7 Eu/kT 8*10 B(T)e Nm h'TfWH(A/F2)V1guAtL iT [14] Theequation for Ai can be somewhat simplified because the thermal noise is generally (35) negligible when compared to the shot effect for most systems and so equation [11] becomes r 1/2 Aip = 2ecBMf (id + ic)] 1/2. 15] Substituting for Aip, ic and Aic and collecting terms, the expression for the minimum detectable concentration is obtained Nm = [16] 3.8 x 1034B(T)eEu/kT(2ecBMcf [id + YTfIcWH(A/F2)s] + [CTf IcWH(A/F2) s 2f)1/2 9oguA LtTfWHA/F2 in units of atoms/cm.3 of flame gases. The constant 3.8 x 1034 results when 8n x 107 is divided by Planck's constant. If Nm in equation [16] is differentiated with respect to W and minimized, then the condition for optimum slit width in cm., Wo, is found to be satisfied when [rT H(A/F IcRd 2Wo4 + [YTfH(A/F2)cI scRdWo3 ecBMYTfH(A/F2)IcscWo 2ecBMid = 0 [17] By evaluating all factors in equation [17] for the particular experi- mental setup in concern, Wo can be found. (Use of a graphical method is the simplest means.) If the flame background is low at the wavelength setting of the monochromator, then sm is much larger than s and s = sm = RdW. For this case W is given by an expression considerably simpler than equation [17], namely, .2ecBMid 1/4 Wo ( [TfTcH (A/F2)Rd2 [18] where equation [18] can be found directly from equation [17] by assuming s is very small so that the second and third terms become negligible. As Ic and AIc increase (AIc increases approximately proportionally with Ic (17)), the last term in equation [17] becomes negligible when com- pared to the other terms. Therefore, at large values of Ic, the optimum slit width, Wo, decreases indefinitely and approaches zero. The use of concentrations in atoms per cm.3 of flame gases would have little meaning to most chemists. It would be more convenient to use concentrations in moles per liter of solution introduced into the given flame. If the concentration of sample is C moles per liter and if the sample is introduced into a given flame at a rate of 0 cm.3/min. or 0/60 cm.3/sec., than 0/60 x 1000 liters or CO/60 x 1000 moles of sample are introduced each second. The number of atoms per second intro- duced into the flame is then C06 x 1023/60 x 1000. If Q is the flow rate of unburned gases in cm.3/sec. introduced at room temperature and one atmosphere pressure, then C06 x 1023/60 x 103 Q is the number of atoms per cm.3 of flame gases. This expression assumes thorough mixing of the sample with the flame gases, no entrainment of atmosphere, no expansion of burnt gases, 100 per cent efficiency of sample introduction (42), and complete dis- sociation of the salt crystals into atoms. With a total consumption atomizer-burner nearly complete mixing of sample with flame gases occurs in the outer cone of the flame. No detailed equations can be given to consider the extent of atmosphere entrainment because it depends on the exact experimental arrangement. The expansion of burnt gases has been accounted for by Alkemade (1) by use of an expansion factor ef. Winefordner, Mansfield,and Vickers (42) have accounted for incomplete dispersion of sample solution into droplets and incomplete solvent evaporation by use of a sample introduction efficiency factor e. A factor P will be used to account for incomplete dissociation of salt crystals into atoms and atomic losses due to ionization and compound formation with flame product gases. Therefore, using ef, e,and P in the previous equation, the flame gas concentration N in atoms per cm.3 of flame gases can be converted to solution concentration C in moles per liter by 6 x 104. efQN C 6 x 1023. [19] The term ef has been evaluated by Alkemade (1) and is given by ef = nTT/n298 298 [20] where T is the flame temperature in oK., nT is the number of moles of combustion products at temperature T and n298 is the number of moles of species at room temperature, which is taken as 2980K. in this case. Thus the final form of the conversion equation is given by C = 3.3 x 10-22 nTQTNer. 21 moles/liter. [21] n298 6 In the above equation, the values of nT and n298 must include not only the moles of flame gas products due to the fuel combining with oxygen but also the moles due to the vaporization of the solvent intro- duced. Equation [21] can be used to calculate the minimum detectable solution concentration of a given atomic species once the minimum de- tectable concentration of atoms in the flame has been calculated. Discussion and Calculations Equation [16] for Nm is a general expression and should allow the accurate calculation of limiting detectable concentrations for any element, present in any given flame, and analyzed using any given experimental setup, as long as accurate data are available to evaluate the factors. In many cases good data will not be available, and this will prevent accurate absolute values of N from being calculated. How- m ever, it should be stressed that the usefulness of equation [16] lies not only in the calculation of detection limits but also in the pre- diction of the effect of changing various experimental factors on the limit of detection. In addition, if sufficiently accurate data are available, then it should be possible to determine accurate transition probabilities if accurate measurements of absolute intensities are made. It should be noted that equation [16] for Nm is exact as long as the spectral line is single, sharp, and isolated, and if this is not the case, then the other spectral lines passed through the exit slit must be corrected for in the manner discussed in the preceding portion of the paper. Also the equations derived should apply whether single stage phototubes or multistage phototubes (photomultipliers) are used as detectors. The use of equation [17] for calculating Wo and equation [16] for calculating Nm is illustrated by the representative results in Table 1 for Na 5890 A. and Cd 2288 A. lines analyzed using the experimental conditions and data listed at the end of Table 1. However, equation [18] gives essentially the same value of Wo as equation [17] because of the low flame background. The values of F, A, Tf, H, and Rd listed in Table 1 were repre- sentative values taken from the manufacturer's literature. Photo- multiplier tube characteristics vary with individual tubes and for accurate work id, M, and ( must be experimentally measured for the particular detector. For the purpose of this paper the value of id given by the manufacturer for a good 1P28 photomultiplier has been taken. Because the amplification factor, M, is related to the anodic photosensitivity, j, by a constant characteristic of the photocathodic surface and because the anodic dark current id equals M times the cathodic dark current, the value of N is independent of M and T. m However, characteristics of phototubes are generally reported in terms of anodic rather than cathodic output, and therefore, anodic dark current and anodic photosensitivity are used in the theory of this manuscript. The values of M and Y given in the manufacturer's litera- ture have been taken as typical. The values of Y at the wavelengths of interest were corrected for variation in spectral response from the manufacturer's response curves. The gain, G, per stage of a photo- multiplier tube is of the order of 4, and so the value of B is taken as 1.3. The absolute background intensities, Ic for oxyhydrogen and oxyacetylene flames in units of watts/cm.2 ster. mp have been taken from the plots of Ic versus wavelength given by Gilbert (17). Gilbert also gives similar plots for air-hydrogen and air-acetylene flames. The value of AIc is given by cI,, where y is the ratio of the root- mean-square fluctuation in the background intensity to the value of the background intensity. The value of f depends upon the temperature fluctuations, irregularities of atomization at the spray tip, mechanical turbulence in the spray and the burning gases, and turbulence due to friction and mixing with the ambient air (17,20). Of these factors only od a SS F - O -4 0 OO U 0 0 0 W HO prz 0 C 0 Q a0 a 0 c eO M H4 H co 0 a So > C E-4< B1 0 Q) 0 0 *0 0 , I oe e C 0 CM4 U U 0 0 u S 0 0 0 u p z 4J Q)o .0 C4 0 4-) a n Z 0'1 0 -"H CO 0 O 0 4J U, 0 0 - o a 0 -4 3 o o rI 4 0 r-4 CO C4 cu 0 Le -4 -I 0 0 0 a -4 co -4 \o 0 O 0 0 -4 -4 0 4-4 o 0 zo * .N -41 0 -r4 41 O 0 :3 Cd r-44 -I 0 a bo 0l 0 0 0 -4 ,O :J -4 u r-- -I 0 .1 0 o5 0 0f i- 0 o 0 -4 M 0 a- a0 -4 a41 rl0 il 0 *U 1- M 00 4J CO, N a- 0 0u OI co 0 0O 0 u -4 II a 0 C--' II 0 6 0 I fto *Il a * a) L0 0 -4 01 o41 II C0 I0 : C 4- 1 0 ca .> Q1) -,C cc4 -4 0 0 4-) u --* c0 CO 0 0c -4 H 0 U II a p . CO r4 0 (0 411 0 to. 0 00 (d a m a o ico o o 0 II -1 a * 00 0 a-I C5 i-0 01 0-cd > r~cc 40 a 0 -14 4-4 n ) X S- 0 Q) 00 o in -H* ai I-ca4 0 ) C a r4- 0 ca CN cc Cu C) 00 0a 41 4o o 41 p3 0 0 cc 0 -40 0 0 a c 5i C C Tl C Ue 1 i 0 4J a .0 0- rl me S0 cda 00 &4 oa 0 oo co u * 4 ^ cc 4 0 M 0 0 a -4 M . 0 u 0 0 0 0 () 0 Fj 04( OH ca s-4 4 60 0 .- ,* 0 0 -4a a On .o (3 * 4 -0 0 0 0 pH 0 0 cc M 00 0 0 Sed 4 -rt * co 4 c-- -0 0 c' 5 -1 co in *d c* I O 0 0 * M 0 CM00 C) H ul a II 0 in o 0 0 0c a o o 0 0 Co 0 4,a -4 4 -4 0 aO 4o 0 4J vu a,0 -d rl 01 03 ai 0 0 -4 01 0) i-- -4 4- 0 0 co CM 0 4-i 0 0 a i4 d II O H--1 II II -~4 00 * v-40 I o co II 0 * IM .4 T- 0 U II 00a O . 0 c- SII a 0 O4 0 II O -H the fluctuation due to temperature variation can be readily represented by an expression. The background intensity of the continuum varies approximately as exp (-hV/kT) and so an approximate value of can be found by differentiating Ic with respect to T to find the relationship between I and AT, the root-mean-square temperature fluctuation. If this is done, is found (1) to be given by (hV/k) (FT/T2), which will be smaller than the true due to neglect of the other factors mentioned above. A Beckman flame in good adjustment (at 30000K. and 5000 A.) results in a value of of approximately 0.005. If only temperature variation is considered, this value of will result from a temperature fluctuation of about 20K. The use of a sheathed burner as described by Gilbert (18) allows even better stability. Experimental values of can be measured by finding the magnitude of the peak-to-peak noise as compared to the value of the background signal and multiplying by 42 /4 to convert to r.m.s. noise (19). By the use of equation [21], the Nm values in Table 1 can be con- verted to more meaningful solution concentrations in moles/liter. For a typical oxyacetylene flame with a temperature of 28500K. at 1 cm. above the inner cone (43), when an aqueous solution is introduced into the flame via a total-consumption atomizer-burner, at a flow rate of 2 cm.3/min., a concentration of 1.8 x 106 atoms of Na per cm.3 of flame gases was found to be equivalent to a solution concentration of 1.6 x 10-4 p.p.m. This value agrees quite well with the value given by Gilbert (17). In this calculation the flow rates of C2H2 and 02 were, respec- tively, taken as 2500 cm.3/min. and 3750 cm.3/min. This would result in a value of Q of 125 cm.3/sec. if allowance is made for the sample vapor, assuming that the sample vapor is not appreciably dissociated (46). The data for nT and n298 were taken from the thesis by Zaer (46). The ratio nT/n298 is approximately 1.2. The efficiency of atomization was taken as approximately 0.5 according to data by Winefordner, Mansfield, and Vickers (42). The value of P was determined primarily by ionization of sodium atoms, although the formation of NaOH in the flame (16,24) may cause P to be slightly less than by ionization alone. An approximate value of P-was determined in the following way. The ionization constant for the process Na -- Na+ + e- was calculated to be 1.01 x 10-7 atm. at 28500K. from the expression given by Mavrodineanu and Boiteux (30). The partial pressure of electrons due only to the flame gases was taken as approximately 4 x 10-9 atm. (15) in the outer cone of a C2H2/02 flame at 28500K. This partial pressure of free electrons corresponds to 1010 electrons per cm.3 of flame gases assuming the flame gas solution of free electrons is an ideal gas. Using the above data, the degree of ionization was found to be 0.96 and, therefore, P was found to be 0.04. The degree of ionization calculated agrees quite well with values cal- culated by Gaydon and Wolfhard (15) and Foster (12) for the outer cone of acetylene flames. The above calculation did not account for additional dilution of the analyte vapor due to entrainment of ambient air. Except under accurately controlled conditions, this additional effect is not deter- minable. If this effect is accurately known, then Q and nT/n298 should be correspondingly corrected. However, this is unnecessary unless the most accurate possible results are needed, such as when the described theory is applied to the calculation of transition probabilities for unknown cases. If this is done, it is best to use well defined, sheathed., homogeneous flames in order that Nm is accurately known. In Table 2 the effect of flame temperature on Wo and Nm is given for the resonance lines of Na and Cd when analyzed by a Beckman DU mono- chromator IP28 photomultiplier detector combination. The resonance lines of Na and Cd differ quite greatly in wavelength and in excitation energy and were especially chosen to illustrate the influence of excitation energy on the limit of detectability as a function of several flame temperatures. No single flame type can be used to cover the broad range of flame temperatures listed in Table 2 (see item 2 at end of table), and so the flame type is not specified. However, a stoichio- metric air-acetylene flame has approximately a temperature of 20000K., a stoichiometric oxyacetylene flame has approximately a temperature of 30000K., and an oxycyanogen flame has approximately a temperature of 40000K. when aqueous solutions are introduced from total consumption atomizer-burners at the rate of about 1 cm.3/min. The values of Ic listed in Table 2 were considered typical and should not be considered as accurate values for the above flames. As can be noted from Table 2, an increase in flame temperature results in an improvement in limit of detectability and a decrease in optimum slit width. With elements having their resonance lines in the u.v., an increase in flame temperature gives a great increase in sensi- tivity as a result of the exponential factor in equation [16]. How- ever, for elements with their resonance lines in the visible the increase 1- On 0 I 0 0 0 o i- n $ -4 ,-4 :j u u),4 0 4-1 Q> (U) a :3 00 1 M 0 0o> C a W up u o co u-4 -M < C ud l E-4 c -4 0 O C00 00 P 0 n 0 0 0-00 0 *4 0- .a O 0 ) 4 40 H 0 oO 0 r- 0 . E-4 0 -4 0 Q r HZ 0O- 0 0 0 , ZOO 0> 4 0 0 C 4 4 4- - > 00 0 >-3 E0l l 4l 44-4 -- 4 04 >iO 0 0 0 p-3 [.0 U .* 0 0 4 HOM 0 S E- 044 ( 00 4 Q 4 r -400 E 0 00 4- 0 40 4 0 O-4 ,- 0 l "4 l m)J -4) a H -4 OOHO U.r 4p 4a w0 o -4 0 4l O l 6 up y oo r 4) p a u "l a) 4 Q l [-.U 0 (0 0 ,- 0 4J-4 0 41 4 q OP14 0 4 r00 0 0 I 4 0 a) a 4 0H 0 0 00 0 *0 O 30 i (1 0 00 0 O -- *-4 l 00 0 -0 > lt 4 C H t H Cu 0OU-) u -4r 0 - Q<) 0 l O 0 X4 -,l cU SH C *- 4m) 41 t 00 0 (1 Z Z-i o N XN^ 41 0 >% 0's C 0u *41 CU z cu po -a C ca : M 0 l Ml X ) 20 o 0 -o40 4 l0 4 41 0.0Q 4.Ha H4JO O 0 0 C -) \< 17 3 "m CNO i 0) gl f r P. i 00 0 0 C O H 0 p l'-4 0 0 w '-4ue4- ") 00 l uO 0 % 0 C0 O .4 4J 0 0 4 r- 0 3 U0 0 r 0 *4 x x U 0 H 40 .E-D (U) 41 4 C0 e H 0 C :.% C. l O CIJ-r:. 0 O 3c 0 E '0 *2 o > *6 o0 E-4 W c4 -L4 O 4J .4 *i0 u co p S a CO 44 U ) (. 4 -',4 *O 0 SE- 0 ( 0 C )E-4 0 o C1 r4 0 a 3 4 0o ZM r r=: 0 No Cl.1 U) M M 4J PQ Ca u Z >*- 4 C 0 40 1* 41 1 0 4 a0 00 0 w- Ca 1 U 3 C 4 0 0 * CE-I ) C u- /3 U 4 a e bO -4 3 0b14 0-0 0 B -.-4 -Li O W -0,4 O kD 0 a : 0) 404 () 0 0 aw 44 0m o *4 > > HO0M 0 w 0A w E- > 0 O- I *- E4 W4w 0 a) 13 4 4 () 04 q 0 (O VI 0 C : 0) 0 mU 01 3 E, ( e *O uo 0 n c4 rA u o I > 0 o () r-4 ( O 4 o u 40 t0 Ht 41 a V 4 0 pQ 0 0 a 0 - 44 z< u 0- o J O in sensitivity, i.e., decrease in value of Nm, with flame temperature is not as great as might be expected. This neglects any adverse effects due to ionization and is due only to the increase in Ic and aIc asso- ciated with an increase in flame temperature and a smaller increase in the exponential factor of equation [16]. Therefore, for elements with spectral lines in the visible, no significant advantage may result by the use of higher temperature flames, e.g., oxycyanogen or fluorine-hydrogen, rather than the usual flames of oxyhydrogen or oxyacetylene (or even air- acetylene), and in some cases an exceptionally high background may result in a decreased sensitivity. The data used to obtain the results given in Table 2 are listed at the end of the table. The shape of the Nm versus slit width, W, curve for any particular experimental setup and spectral line should be of special interest to the analyst. If this curve has a broad flat minimum, then the exact setting of the slit width for optimum conditions is not critical. Because the shape of the Nm versus W function is independent of the spectral line, it is not necessary to evaluate all factors in equation [16] at a variety of slit widths, but rather it is possible to combine all factors except 1/W outside the square root sign into a constant factor designated Kc. This constant Kc is a function of the particular line, flame and instrumental setup but is independent of W. Therefore, by replacing s by RdW and evaluating all universal constants, equation [16] can be rewritten as Nm = [22] (Kc/W)(5 x 10-19M[id + TfIcH(A/F2)RdW2] + [YTf1cH(A/F2)Rd]2W4)1/2 where all factors within the square root must be evaluated for the particular experimental conditions. It is evident from the above equation that the shape of Nm as a function of W is not dependent on the magnitude of K In Table 3 values of Nm have been calculated from equation [22] for various W values, and, as expected, the minimum value of N occurs at Wo. As can be seen from the results in Table 3, Nm varies approximately linearly with W for large W values and 1/W for small W values. The particular experimental conditions for which the calculations were made are given at the end of the table. For the experimental conditions listed in Table 3, the Nm versus W curve has quite a flat minimum. In fact the value of W can deviate from Wo by a factor of 2, and Nm in- creases by a factor of no more than 1.4. The above discussion regarding equation [22] is valid only if the spectral line is resolved as a single, sharp line. If two or more spectral lines are collected by the exit slit, then equation [22] must be modified accordingly, as previously discussed. In the latter case, a plot of Nm versus W may result in more than one minimum of Nm for certain optimal wavelength settings. For example, with hydrogen flames and prism monochromators, the most sensitive spectral line of elements such as Fe, Co, Ni, Ru, Rh, and Cr are not fully separated. Even the best prism monochromator with the narrowest useable slits may not be able to isolate the sensitive lines of the rare earths. Several other cases will be given which demonstrate the effect of various factors on the optimum slit width. A small value of Wo (Wo = 0.002 cm.) results when using equation [18] for the following VARIATION OF TABLE 3 Nm WITH W FOR A PRISM MONOCHROMATOR AT 420 mp WHEN USING AN OXYHYDROGEN FLAME* W, cm. Nm 0.001 0.005 0.008 0.010 0.012 0.015 (Wo) 0.020 0.030 0.050 0.100 7.1 x 10-8K 1.5 x 10-8Kc 1.0 x 10-8Kc 0.90 x 10-8Kc 0.82 x 10-8Kc 0.78 x 10-8Kc 0.79 x 10-8Kc 0.91 x 10-8K 1.3 x 10-8Kc 2.6 x 10-8K *The above calculations were performed using equation [22] and the following data: id = 10-8 amp. (for a good 1P28 photomulti lier tube), = 104 amp./watt, M = 106, Tf = 0.5, H = 1 cm., A = 25 cm. F = 50 cm., Rd = 100 2m/cm., Af 1 sec.-1, and AIc = 5 x 10-11 watts/cm.2 ster. mp sec. Using the above data and equation [18], Wo is 0.015 cm. conditions: id = 10-8 amperes (a good 1P28), W = 104 (a good 1P28 at 400 mp), B = 1.3, M = 106, Tf = 0.5, Rd 100 mp/cm. (for a prism mono- chromator at 400 mp), AIc = 3 x 10-9 for an oxyacetylene flame at 400 mp, and HA/F2 = 0.01. For this case the value of W is of the same order as the practical (effective) resolving power expressed in terms of slit width. In the Beckman DU monochromator, the practical resolving power slit width (20) is about 0.003 cm. in the visible and 0.006 cm. at the shortest useable wavelengths, and it is determined by the combination of diffraction, coma, aberrations, slit curvature mismatch, and optical imperfections. Therefore, for this particular case, the limit of detectability will be considerably lowered if the slit width is made smaller than the effective resolving power slit width although no gain in resolution will occur. If in the above case the values of Ic and AIc are increased by 103 corresponding to a brilliant background in a plasma flame, Wo is approximately 0.00007 cm., which is considerably less than the effective resolving power slit width. A case in which a wide slit width value of Wo results would be the case in which id 3 x 10-8 (maximal specification for a 16PMI photomultiplier tube), Y= 102 (at 630 mp), B = 1.3, M = 105, Tf = 0.2 (for a grating monochromator), Rd = 10 mp/cm., HA/F2 = 0.005, and AIc = 3 x 10-12 watts/cm.2 ster. mp sec.-1/2 (a good stable H2/air flame at 630 mp). For this case Wo = 3.4 cm., which is a very wide slit. Assuming no lines are present other than the line of interest, it is interesting to find how much sensitivity is lost if a narrower slit is used. From equation [22], Nm is 1.4 times greater at W = 1.6 cm. than at W = 3.4 cm. and is 4 times greater at W = 0.5 cm. than at W = 3.4 cm. 26 In this case the value of Wo found by using equation [18] is so large that it is essentially the same as that given by equation [17]. III. CALCULATION OF THE LIMIT OF DETECTABILITY FOR ATOMIC ABSORPTION FLAME SPECTROMETRY Introduction The system under consideration in the development of this theory consists of a hollow cathode discharge tube (HCDT), or other resonance lamp which emits sufficiently narrow lines, a flame into which the sample is aspirated, a lens for focusing radiation from the source onto the central part of the outer cone of the flame (which can be considered to be approximately in thermal equilibrium), a second lens for focusing the transmitted radiation onto the monochromator entrance slit, a mono- chromator with accompanying optics and slits, a photodetector, amplifier, and readout (recorder, meter, etc.). The HCDT emits a line of intensity 10 (in watts/cm.2 ster.), and the intensity of the transmitted radiation is It (in the same units as 10). The transmitted radiation is focused by the second lens onto the entrance slit of the monochromator so that the slit is fully illuminated and the effective aperture of the spectro- metric system is filled with radiation. It is assumed that correction is made for thermally emitted and fluorescent radiation of the same frequency as the incident radiation so that It is the true transmitted intensity, i.e., it does not include emission from the flame other than flame noise in the frequency interval, af, over which the amplification-readout system responds. Also it is assumed that correction is made for any radiation loss due to scattering by droplets in the flame. Further it is assumed that the monochromator is set on the line center, yo (the frequency of the absorption line peak in sec.'1), and that only radiation of the line of interest enters the spectrometric system. Derivation of Equations If the spectral line emitted by the source is much narrower than the absorption lines in the flame, then 10 is related to It by Beer's Law A = log (l1O/t) = 0.43koL [23] where Ao is the absorbance, ko is the atomic absorption coefficient in cm."- at the line center for any atomic concentration, and L is the length in cm. of flame gases through which the radiation is passed. Justification for the assumption of a narrow source line is to be found in the statements of Jones and Walsh (26) and Crosswhite (11) and in the ability of the experimenter to adjust the source line width by changing the operating current of the tube, or, if necessary, even by cooling the tube. (However, see Appendix C for a further discussion of the effect of line width.) Thus it is seldom necessary to operate under conditions in which the effect of Doppler, Lorentz, and Holtsmark line broadening is more important in the source than in the flame. However, it may occur that a line is intrinsically broad due to hyperfine structure, both from nuclear spin effect and isotope shift. It may be shown (see Appendix D) that the effect of hyperfine structure is to reduce the absorbance below the expected value. Thus A = log (IO/It) = 0.43pkoL [24] where p is the correction factor to account for the decrease in absorbance. The method of calculating p is indicated in Appendix D. The absorbance will be said to be just detectable when the difference between the detector output signal anodicc current of the photodetector) i, in amperes, due to I1 and the detector output signal it, in amperes, due to It equals twice the total root-mean-square noise fluctuation. It is assumed in this theory that the noise is measured at the wavelength of the line of interest with solvent aspirated into the flame. Thus the minimum detectable signal difference is given by (io it)m = 2iT [25] where AiT is the total root-mean-square noise fluctuation in the output current of the photodetector in amperes, and the subscript m indicates that the signal difference is that observed at the minimum detectable concentration. The relationship between signal in amperes and intensity 2 in watts/cm.2 ster. has been derived in Appendix A. Thus it may be shown that i Y-TfWH(A/F2)Io [26] and it = TTfWH(A/F2)It [27] where all the terms have been previously defined. Substituting for i and it in equation [25] one obtains YTfWH(A/F2)(1 It)m 2 [28] However, from equation [24], It can be written in terms of 1 to obtain YTfWH(A/F2)I(1 e- kmL) = 2A [29] where ko is the atomic absorption coefficient at the line center at the minimum detectable concentration. Expanding the exponential term and noting that komL is small at the limit of detection, i.e., e-pkOmL 1 -pkmL, one may write TfWH(A/F2)IokomL = 2ZiT [30] The root-mean-square noise signal, AiT, may be attributed to fluctuations originating in the photodetector, Aip, fluctuations in the source, Ai, and fluctuations in the flame continue, Aic. The noise signals add quadratically (18), and therefore, AiT is given by -022 -- 2)1/2 AiT (Aip + o + i2 2 [31] The value of Aip is given (35) by 6ip = [2BMecAf(id + i + i)]1 [32] where B is a constant approximately equal to 1 + 1/G + 1/G + ..., where G is the gain per stage of the photodetector, M is the total amplification factor of the photodetector, ec is the electronic charge in coulombs, id is the photodetector dark current in amperes, and ic is the anodic current of the photodetector in amperes due to the continuous emission of the flame. In any practical case in atomic absorption flame spectrometry id and ic will be much smaller than 1i or, at least, can be made much smaller at the limit of detectability. Hence one may write Aip (2BMecAfi)/2 [33] and substituting from equation [26] one obtains for 2i i-p2 = 2BMecAfXTfWH(A/F2)Io [34] The value of Ai2 is related to the fluctuation in the intensity of incident radiation by o2 = [YTfWH(A/F2)To]2Af [35] where Al- is the fluctuation in the intensity of radiation from the source in units of watts/cm.2 ster. sec.-/2 The sec.-1/2 unit arises because the noise is a root-mean-square noise. The relationship between the noise signal and the response bandwidth, Af, in sec.-1, has been more fully discussed in Section II. Similarly, Aic is given by Aic = [TfWH(A/F )Acs ] Af [36] where AIc is the fluctuation in the intensity of the continuum emitted by the flame in watts/cm.2 ster. mp sec.-1/2, and s is the effective spectral slit width (in mp) of the spectrometric system. The spectral slit width is given by the expression s (m2 + sc2 1/2 [37 where sm is the spectral slit width in mp as determined by the mechanical slit width, and sc is the spectral slit width in mg at an infinitely narrow mechanical slit width as determined by diffraction and by coma, aberrations, mismatch of slit curvature, imperfection of optics, and other factors characteristic of an imperfect monochromator. Expression [37] differs from the one given for s in Section II since in the atomic absorption case, the more exact equation for s can be used without overly complicating the theory. It will be convenient in calculating the noise terms to write the fluctuation as fractions of the total intensity such that AIo XIo and AIc Ic, where X and f are the appropriate fractions, and Ic is the intensity of the continue emitted by the flame in units of watts/cm.2 ster. mp. Equations [35] and [36] may then be rewritten in the following form: A02 [YTfWH(A/F2)xIO]2Af [38] Ai2 = [YTfWH(A/F2 )Ics ]2l f [39] Substituting the appropriate values into equation [31] one obtains for AiT i--T = 2BMecIo 2 1 /2 Ai = [YTfWH(A/F2)]2 ( YTfWHA/F2 + (X0) + (cs)2 [40] The value of kom, the atomic absorption coefficient at the line center at the minimum detectable ground state concentration Nom, is given (31) by o 2 ln o2 gu NomAtf 0.037 o2guNomA [41] km = ui, [41] AV D ~8t8go go0VD where A'D is the Doppler half width (in sec.- ) of the absorption line, Xo is the wavelength (in cm.) of the line center, gu and go are the statistical weights of the upper u, state and the lower (ground, o) state, At is the transition probability in sec.-I for spontaneous 'emission, i.e., state u to state o, and S is a factor to account for line broadening other than Doppler broadening and hyperfine structure broadening (see Appendix D). Because analytically important atomic absorption transitions usually involve the ground state, only transi- tions to that state will be considered. The constants have been evalu- ated in the right hand part of equation [41]. The number of ground state atoms per cm.3 of flame gases, N, is related to the total number of atoms in all states per cm.3 of flame gases, N, by the famous Boltzmann equation, namely No = Ngo/B(T) where go is the statistical weight of the lowest (ground) state and B(T) is the partition function of the atom, i.e., B(T) = ZgeEikT where gi is the statistical weight of the state i, Ei is the energy of state i above the ground state, and the summation is over-all states of the atom. For the case of the minimum detectable concentration, Nom = Nmgo/B(T). As previously discussed in Section II, B(T) is approximately given by go except for a few atoms with low lying atomic states, e.g., Cr. In this paper B(T) for the examples given is accurately expressed by go and so Nm is essentially the same as N The above theory is still correct for transitions occurring to levels higher than the ground state if go is replaced by the statistical weight of the state involved. Substituting for kom and AiT in equation [30], and solving for Nm one obtains 53.5 AVDB(T) 2BMec 2 + ) 1/2 [ pNm LX, TWAT2) + 2Tf ( s'/I) AfguA [42] LAo28guAti TyWH(A/F2 o The above equation was derived assuming that the output photodetector signal due to thermal emission, fluorescent emission of the resonance line,and incident light scattering was either negligible or corrected for. In Appendix E equations for Nm are derived for the case in which compensation is not made for thermal emission for several detection systems. The influence of fluorescent emission and light scattering are also considered in Appendix E. The optimum slit width, Wo, can be obtained by minimizing equation [42] with respect to W, i.e., equating dNm/dW to zero and solving for W. If s is given by s = (RdW)2 + Sc2 1/2 from equations [2] and [37], and if sc is assumed constant with change in slit width, then Wo, in cm., is given by ( BMecIo 1/3 Wo = yTfH(A/F2)(qIcRd)2) [43] The minimum detectable concentrations in atoms/cm.3 of flame gases can be converted to solution concentrations in moles/liter by use of the equation derived in Section II (equation [21]). Discussion and Calculations When one attempts to calculate minimum detectable concentrations through the use of equations [42], [43], and [21], the need for further measurements of basic experimental parameters of atomic absorption flame spectrometry becomes apparent. One of the most important contributions of these equations is that they point out those areas most in need of further work. It is difficult to compare calculated limits of detectability with measured limits of detectability unless accurate absolute values of 1, Ic, X, Y and the photomultiplier tube characteristics are stated. Also, more information is needed on I values and on the degree of compound dissociation and atomic ionization so that P may be calculated with some accuracy. Although the calculation of limits of detectability is at present hampered by this lack of information, equation [42] is still valuable in that it allows one to calculate the effect of varying one or more of the experimental parameters on the sensitivity of analysis, e.g., the effect of slit width or temperature on Nm. Table 4 shows the results of calculations of the minimum detectable concentrations for two favorable cases. When one chooses a good instru- ment such as the Beckman DU or the Jarrell-Ash 500 mm. grating mono- chromator, a 1P28 photomultiplier detector, and flame conditions such as stoichiometric oxyhydrogen or oxyacetylene, then, assuming a reasonable value for 10 and using the optimum slit width, all the terms under the square root in equation [42] except X2 are negligible. It is assumed that correction has been made for the output signal of the detector due to thermal and fluorescent emission and light scattering. Thus for these conditions, which are of experimental importance, the value of N de- pends directly on X the fluctuation of the source intensity, and so equation [42] can be reduced to 53.5AV DB(T) X /2 Nm = .pI[2guAt [44] Equation [44] is in agreement with the expectation that over a fairly broad range of experimental conditions, the sensitivity of atomic absorption flame spectrometry would be independent of instrumental parameters. For the case represented by equation [44] the limit of TABLE 4 REPRESENTATIVE RESULTS IN ATOMIC ABSORPTION FLAME SPECTROMETRY FOR LIMITS OF DETECTABILITY IN ATOMS PER CM.3 OF FLAME GASES FOR Na and Cd IN TWO FLAMESa Flame Temp. OK. Atomic Line 2700b 2850c Na 5890 A.d 1.1 x 1010 5.6 x 109 Cd 2288 A.e 4.3 x 1010 2.2 x 1010 aX = 10-2, Af bStoichiometric 2 cm.3/min., L = 1. H2/02 flame with aqueous solution flow rate of about = 0.5 cm. CStoichiometric C2H2/02 flame with aqueous solution flow rate of about 2 cm.3/min., L = 1 cm. dAVD = 4.1 x 109 sec.-1 at 27000K., 4.2 B(T) g=^=2, g = 4; p = 0.97; Xo 1.3 x 108 sec.-l;u = 0.43. x 109 sec.-1 at = 5.89 x 10-5 eVD = 4.6 x 109 sec.-1 at 27000K., 4.7 x 109 sec.-1 at B(T) go = gu = 3; ( 1; X = 2.29 x 10-5 1.7 x 108 sec.-1; u 0.43 28500K.; cm.; A = 28500K.; cm.; At detectability could be greatly improved only by an increase in path length (13) or an increase in the stability of the source. However, from equation [42] it may be seen that as the stability of the source is improved, other terms might become significant. The term (FIcs/Io)2 becomes important as the flame background-increases, e.g., with the highly fuel rich flames frequently used with elements that form very stable oxides, or as the spectral slit width increases, e.g., with filter instruments. However, for the conditions listed under Table 4, the minimum detectable concentration may be calculated from equation [44]. The values of the parameters given in Table 4 were found as follows. The Doppler half width, AVD, can be obtained from the expression given by Mitchell and Zemansky (31) AVD 2 2Rn 2 T( ) 1/2 cMa 7.15 x 10-7 ~o0 ) [45] where R is the gas constant in ergs/mole. oK., c is the speed of light in cm./sec., T is the absolute temperature, and Ma is the atomic weight in atomic mass units. Universal constants have been evaluated in the right hand part of equation [45]. Values of 9 in terms of the damping constant a = (6VL/AVD) Jin 2 (where aVL is the Lorentz half width in sec.-1) have been tabulated by Poesner (33). The value of a for Na 5890 A. was obtained from Hinnov and Kohn (22). Although the value of a for Cd 2288 A. is not readily available, its value is estimated to be about the same as that given for Ag 3281 A. by Hinnov and Kohn (22). Other listings of a values are to be found in James and Sugden (23), Sobolev (37) and Mitchell and Zemansky (31). A fuller discussion of the shape of spectral lines and its effect on atomic absorption is con- tained in Appendix C. The calculation of p is discussed in Appendix D. The results given in Table 4 in atoms/cm.3 can be converted to solution concentration by the use of equation [21]. A minimum detect- able concentration of 5.6 x 109 atoms/cm.3 of flame gases (for Na 5890 A. at 28500K. (43) and the conditions given in Table 4) corresponds to a solution concentration of about 0.5 p.p.m. The terms in equation [21] have been taken as follows: Q = 125 cm.3/sec., nT/n298 = 1.2, 0 = 2 cm.3/min., 6 = 0.5, and = 0.04. If the flame cell length were increased to 10 cm. the limit of detectability would drop to 0.05 p.p.m. and if the cell length were 100 cm., the limit would be 0.005 p.p.m. (13). Flames in which compound formation and ionization are less, will result in a value of P greater than 0.04 and, therefore, a lower limit of detectability. The limit of detectability calculated for a 10 cm. long flame compares quite favorably with measured values in the literature (3,14) for slightly different flame conditions. In Table 5 the shape of the Nm versus slit width, W, curve is illustrated for several sets of conditions. Because the shape of the Nm versus W curve is independent of the spectral line, it is not necessary to evaluate all factors in equation [42]. The terms outside the square root have been combined in a constant term Ks, and equation [42] has been rewritten in the form ( 2BMecLf f Nm = Ks TfWH(A/F2)Io + 2f + (cRd/io)2W2) 1/2 [46] where s has been replaced by RdW from equation [2 ]. VARIATION Monochromator Flame Type W (cm.) 0.02 0.2 2.0 20 200 TABLE 5 OF Nm WITH W FOR SEVERAL MONOCHROMATORS AND FOR SEVERAL FLAME TYPESa Jarrell-Ashb DUC Dud H2/02e 10-2 10-2 10-2 10-2 10-2 C2H2/02f 10-2 10-2 10-2 10-2 1.2 x 10-2 H2/02 10-2 10-2 10-2 10-2 10-2 C2H2/02 10-2 10-2 10-2 1.1 x 10-2 4.5 x 10-2 H2/02g 10-2 10-2 10-2 10-2 1.1 x 10-2 aB = 1.5; M = 106; e 1.6 x 10-19 could ; 1 = 8.9 x 10-4 watts/cm.2 ster.; Y= 5.3 x 10 amps./watt at 400 mp, 1.2 x 104 amps./watt at 600 mp; Tf = 0.5; H = 1 cm.; A = 25 cm.2; F = 50 cm.; X= 10-2; = 0.005; s = RdW. bRd = 16 mp/cm. CRd 100 mp/cm. at 400 mp. dRd = 350 mp/cm. at 600 mp. ec = 1.3 x 10-8 watts/cm.2 ster. mp at 400 mp. flc = 4.0 x 10-7 watts/cm.2 ster. mp at 400 mp. gIc = 1.0 x 10-8 watts/cm.2 ster. mp at 600 mp. hic = 6.8 x 10-8 watts/cm.2 ster. mp at 600 mp. (All Nm values in the above table have been divided by the constant factor Ks.) C2H2/02h 10-2 10-2 10-2 10-2 2.8 x 10-2 b In these calculations typical values of B, M, and Y are taken for a 1P28 photomultiplier, and 10 is estimated to be approximately the value given by Crosswhite (11) for a typical line of an iron HCDT. The values of X and f have been estimated as shown in the table, and Ic has been taken from the work by Gilbert (17). For the chosen con- ditions, W would have to be unreasonably small before the first term in equation [46] would become significant. For this reason Nm shows no increase as the slit width is decreased below the calculated optimum value (Wo = 0.19 cm.). However,' slit widths below 0.19 cm. will result in a smaller photodetector signal and correspondingly greater chance for electronic noise interference. As the slit width is increased beyond the optimum value, Nm increases very slowly for the conditions chosen, due to the increase in the last term in equation [46], and very wide slits could be used before the increase would become significant. In this case the slit width used would be determined by the mechanical slit width available on the instrument or by the proximity of other spectral lines. In such instances interference filters can be used with excellent results. The difference in results between the Jarrell-Ash and the Beckman DU is due only to the difference in reciprocal linear dispersion in the two instruments. The variation in results with flame type is due only to the variation in flame background intensity. The variation with wavelength is a result of two competing factors. The flame background intensity is lower at 600 mi than at 400 mp by about 6-fold for an oxyacetylene flame, but the reciprocal linear dispersion of the DU is about 3.5 times higher at 600 mp than at 400 mp. The result is a slower increase in Nm at 600 mp than at 400 mp for the oxyacetylene 41 flame. For the oxyhydrogen flame the results are reversed since the flame background intensity is about the same at both wavelengths, and the higher reciprocal linear dispersion leads to a faster increase of Nm with W at 600 my than at 400 mp. The effect on Nm of varying other factors than those discussed above can be determined by examination of equation [42]. In Appendices C, D, and E some of the details of the preceding derivations and assumptions are discussed to a greater extent. IV. CALCULATION OF OPTIMUM CONDITIONS Introduction A large number of factors affect the signal and the noise in both atomic emission and atomic absorption flame spectrometry, as may be seen from the signal and noise expressions presented in Sections II and III. At concentrations other than the minimum detectable, the expressions of interest for atomic emission flame spectrometry are: i = jTfWH(A/F2)nI [47] and AiT = [2BMecAf(id + YTfWH(A/F2)Ics) + (yTfWH(A/F2)AIc5s)2,f]/2 [48] The expressions of interest for atomic absorption flame spectrometry are: io it = rTfWH(A/F2)Io( ek L) [49] and ---T [50] 2 2BMecI --- 2 1/2 [yrTfWH(A/F2) 2f y2BTecIW + A 2 + (Ics)2 + (Is)2 However, from a practical point of view only a few factors are important in the discussion of optimum experimental conditions. The analyst usually begins with a specified monochromator-detector-readout combination, which places such factors as Tf, A, F, and Af beyond his control. The variables which the analyst is most often interested in optimizing are monochromator slit width, W, flame type (e.g., C2H2/02 or H2/air), and flame conditions (gas flow rates and solution flow rate). If it is assumed, as in the preceding sections, that the entrance and exit slit widths are equal, then the relationship of the slit width to the signal and noise is evident from the above equations. The other factors affect the signal and the noise in a number of ways which are not immediately apparent. It may be seen that the flame background intensity Ic (or AIc), which is dependent on flame type, affects the noise, AiT, in both atomic emission and atomic absorption flame spec- trometry. Qualitatively it may be noted that the signal in both atomic emission and atomic absorption flame spectrometry is a function of the atomic concentration of the species of interest, N, in atoms/cm.3 of flame gases, and the atomic concentration is dependent on compound dissociation and ionization, which depends on flame composition and flame temperature. In this section expressions will be derived relating the signal and the noise to flame type and flame conditions. Selection of Optimum Slit Width An expression for the optimum slit width as a function of experi- mental parameters for atomic emission and atomic absorption flame spectrometry may be obtained by maximizing the respective signal-to- noise ratios, i.e., differentiating the signal-to-noise ratio with respect to W, equating to zero, and solving for the optimum slit width, Wo. The signal-to-noise ratio for atomic emission flame spectrometry is obtained by dividing equation [47] (l= 1) by equation [48]. i/AlT = YTfWH(A/F2)nI [51] [2BMecAf(id + YTfWH(A/F2)Ics) + (tTfWH(A/F2)A cs)2f] 1/ The optimum slit width for atomic emission flame spectrometry is most readily obtained by finding the minimum of the noise-to-signal ratio by differentiating the square of the inverse of the above equation, i.e., (<^/i)2 = (klW)-2 [k2(id + k3Ws) + (k4Ws) 2f] [52] where kl = fTfH(A/F2)nI , k2 = 2BMecAf , k3 = 'TfH(A/F2)Ic , k4 = ITfH(A/F2)Ac. If s is given with sufficient accuracy by RdW, where Rd is the reciprocal linear dispersion of the monochromator in mp/cm., then the optimum slit width is found to be given by [ 2BMecid I /4 [53] 0 = (XTfH(A/F2)c(f) /2Rd) [53] This is the same expression as previously dervied for the opti- mum slit width at the limit of detectability expression with respect to W. The case in which s is not given with sufficient accuracy by RdW has also been discussed in Section II. The signal-to-noise ratio for atomic absorption flame spectrometry is obtained by dividing equation [49) by equation [50]. (io it)/AIT [54] YTfWH(A/F2)IO(1 e-PkoL) 2)j2 2eo 2 2 2 1/2 [fyTfWH(A/F2) BfeTWHA/F2 + Go)2 + Acs) + (Als)2] As in the above case for atomic emission, the optimum slit width expression is most readily obtained by minimizing the noise-to-signal ratio by differentiating the square of the inverse of equation [54], i.e., /(io ic)] 2 = [55] (k5W)-2 [ (k6W)2Af [k7W1 + (o)2 + (cs) + (I)2 ] where k5 = 'TfH(A/F2)IO(1 e-kOL) k6 = lTfH(A/F2) , k7 = 2BMecIO/(YTfHA/F2). The exact expression for s may be substituted into equation [55] without overly complicating the results, i.e., s = (Rd22 + sc2) /2. The various expressions for spectral slit width have been more fully dis- cussed in Section II. If the indicated operations are carried out after substituting for s in equation [55], the optimum slit width expression is found to be BMec0o 1/3 o = [ cRd) 2TfHA/F2 [56] As might be expected this is the same expression as previously obtained by minimizing the limit of detection expression for atomic absorption flame spectrometry with respect to slit width. Effect of Flame Conditions on Atomic Concentration The selection of optimum flame conditions is a somewhat more difficult problem than the selection of an optimum slit width. The origin of this difficulty resides in the nature of the flame source. The temperature of the flame source and the composition of the flame gases affect the atomic concentration of the species of interest through compound formation and ionization, but the flame temperature and flame gas composition cannot be chosen independently. There exists for any given flame a relatively narrow range of conditions of gas flow rates and fuel-to-oxygen ratios over which it is a stable, analytically useful flame, and for any given flame there is a relatively narrow temperature range (43) over which it can be adjusted. Therefore, an optimum tem- perature cannot be chosen in the same way that an optimum slit width can, because a major change in flame temperature involves a change in flame type (e.g., from C2H2/02 to H2/02), with corresponding changes in Ic, AIc, and compound formation of the species of interest with flame gas constituents. To find the optimum flame conditions the procedure which will be followed is to choose a particular flame and calculate the variation of the atomic concentration of the species of interest, N, and the variation of the signal-to-noise ratio over the temperature range which that flame can exhibit. The calculation of N and the signal-to-noise ratio then will be repeated for other flames. For both atomic emission and atomic absorption flame spectrometry, plots of signal-to-noise ratio versus temperature for several flame types will allow the selection of the optimum flame type for the particular analysis of interest. The atomic concentration in the flame of the species of interest is, of course, related to the concentration of the solution aspirated into the flame, as well as being related to compound formation and ionization. No derivation of N is meaningful unless N can also be related to the solution concentration C. An expression has been derived in Section II (equation [21]) relating N and C. In slightly altered form this equation reads N/P = 3 x 1021 n298C [57] nTQT In the following derivation it will be more convenient to relate C to PT, the total pressure in atmospheres of the species of interest present in all forms, atomic, molecular, and ionic. The term N/P is the total concentration, NT, in particles/cm.3, of the species of interest in all forms. Because NT is small compared to other species for most flame spectrometric studies, NT may be related to PT by means of the ideal gas expression (30), PT = NTkT, where PT is in atmospheres, k is the Boltzmann constant in cm.3-atm./OK. (1.38 x 10-22), and T is the flame temperature in OK. The desired expression is, therefore, P, 3 x 10k n298GC [58] T QnT For a given value of PT, the equilibrium concentration of atoms of the species of interest present in the flame is dependent upon the dissociation of the aspirated salt, the formation of compounds with the flame gases, and ionization. If equilibrium is assumed, N can be pre- dicted by consideration of the above processes. For the case in which an aqueous solution of sodium chloride is aspirated into the flame, the formation of only one flame gas compound, NaOH need be considered (24). Sodium chloride has been chosen as the aspirated salt because this is probably the most common means of introducing sodium into a flame, although no change in the results of the theory occur if NaN03, NaC104, or other similar salts are considered. Most sodium salts, except NaOH, are completely dissociated at the temperature of flames used in flame spectrometry. The following equilibria expressions and their corresponding material balance expressions may be written for sodium aspirated as sodium chloride. The dissociation of the aspirated salt, NaCl -- Na + Cl , has an equilibrium constant K1 given by K1 = PNaPC1 [59] PNaC1 where PNa' PC1' and pNaCl are the partial pressures in atmospheres of Na, Cl, and NaCI, respectively. Assuming that Cl in the flame comes only from the dissociation of the salt, and that the pressure of HC1 formed in the flame gases is a small fraction of the total pressure of Cl in the flame, and further assuming that PNaC1 < PT, then pCI is given by PC1= PT PNaC1 T [60] When using H2/02, H2/air, and C2H2/02 flames, the above assumptions are valid. The stability of HCI is low (25), and so dissociation (see Appendix F) should be nearly complete. The stability of NaC1, as will be evident, is also low, and so the assumption PT >> PNaCl is also valid. Substituting for pCl in equation [59] and solving for PNaC1 gives PNaC = PNaT [61] Kl For the ionization reaction Na -- Na+ + e , the equilibrium constant, K2, is given by K2 = PNa+Pe [62] PNa where PNa+ is the partial pressure in atmospheres of sodium ions in theflame, and Pe is the partial pressure of electrons in the flame. The partial pressure of electrons in the flame is the sum of the partial pressures of electrons due to ionization of the flame gases and electrons due to ionization of the metal, i.e., e = Pe + PNa+ [63 where Pe is the partial pressure in atmospheres of the flame gas electrons. Substituting for Pe in equation [62] gives K2 = PNa+Pe + PNa +264 2 PN [64 PNa Solving for PNa+ by the quadratic expression gives -Pe + (Pe2 + PNa = 4K2PNa)1/2 [65] For the dissociation reaction of the flame gas compound NaOH, NaOH --- Na + OH , the equilibrium constant K3 is given by K3 = PNaPOH PNaOH where pOH and PNaOH are the partial pressures and NaOH, respectively, present in the flame. PNaOH = PNaPOH K3 Because PT is the total pressure of the sent in all forms, PT is given by in atmospheres of OH Solving for PNaOH gives [67] species of interest pre- PT = PNa + PNa+ + PNaC1 + PNaOH [68] It may appear that the forms NaH and NaO should also be present in equation [68], but it was theoretically verified from dissociation constants of NaH and NaO that even in flames of relatively large PH and pO, the NaH and NaO species are essentially completely dissociated. The influence of H atoms from the flame gases on the formation of HCI in the flame has not been considered because, as will be seen, the [66] dissociation of NaCl is essentially complete, and this effect will not produce any changes in the results of the theory. In Appendix F the effect of introduction of an excess of Cl into the flame is considered i.e., introduction of NaC1 into the flame in the presence of HC1. Sub- stituting for PNaC1' PNa+' and PNaOH from equations [61], [65], and [67], one obtains 2 1/2 PT PNa + -Pe + (Pe + 4K2PNa) PNaPT + PNaPOH [69] 2 KI K3 Clearing fractions and combining terms, one obtains 2(KlK3 + K3PT + KlPOH)PNa + K1K3(Pe2 + 4K2PNa)1/2 KK3(2PT + Pe) = 0 [70] In order to use equation [70], the equilibrium constants K1, K2, and K3 must be known at several temperatures over the temperature range of the flames of interest. The partial pressure of OH, POH, and the partial pressure of flame gas electrons, p must also be known for the flames of interest. These factors, pOH and pe, are characteristic of the particular flames and vary somewhat with temperature. However, in the calculations to follow, pOH and Pe will be assumed to be constant over the temperature range of a particular flame. Solution of equation [70] for pNa for a particular flame and a number of values of PT is somewhat tedious (a graphical method seems to be the easiest), but fortunately two limiting cases which are of practical importance may be noted in which equation [701 can be greatly simplified. In most of the calculations reported in this paper, one or the other of the simplified forms can be used. In the first case, if ionization of the sodium atoms is unimpor- tant, i.e., pe2 >> 4K2PNa, then equation [70] becomes (K1K3 + K3PT + K1POH)PNa KK3PT = 0 [1] and equation [71] is readily solved for pNa KIK3PT PNa = [72] PNa K3 + K3PT + KlPOH [72] The second case occurs when the partial pressure of flame gas electrons becomes negligible when compared to those produced by ioniza- tion of the metal. In this case (4K2PNa)/2>> Pe and equation [70] becomes 1/2 1/2 (K1K3 + 2K3PT + 2KlPOH)PNa + KlK3K2 PNa 2KKPT = 0 [73] Equation [73] is in the form of a quadratic equation in which Na/2 is the variable. Solving by the quadratic expression and squaring gives PNa = [74] F-KK3K2 1/2 + [(KlK3K21/2)2 + 4(K1K3 + K3PT + KIPOH)KlK3PT] 1/2 2 2(KlK3 + K3PT + KIPOH) The partial pressure in atmospheres of Na, PNa, can be converted by the ideal gas expression (30) to N, the atomic concentration of Na 3 in the flame in atoms/cm. i.e., N = PNa/kT , [75] where k is the Boltzmann constant in cm.3-atm./K. Therefore, multi- plying either equation [70], equation [72], or equation [74] by 1/kT allows the calculation of N as a function of temperature for a number of flame types and a number of values of PT (and because PT is related to the solution concentration by equation [51], N can be calculated at any value of solution concentration). The results of the calculation of N are shown in Figure 1, where log N is plotted versus T for seven values of PT and three different flame types. From Figure 1, the effects of compound dissociation and ionization may be seen, as well as how these competing factors change with PT and with flame type. As PT increases the maxima of the curves, i.e., the temperatures at which ionization becomes more important than further compound dissociation, shift to higher temperatures. As p0H decreases, e.g., in going from the C2H2/02 flame to the H2/02 flame, the maxima of the curves are seen to appear at lower temperatures. In all the calculations whose results are shown in Figure 1, the compound NaC1 was found to be completely negligible with respect to NaOH. The results shown in Figure 1 would be exactly the same if other salts, such as NaNO3, and NaC104, with dissociation constants similar to NaC1, were used for the aspirated species. Optimum Flame Conditions for Atomic Emission Flame Spectrometry Figure 1 is very useful for the discussion of the effects of com- pound formation and ionization, but the signal-to-noise ratio is needed for the selection of optimum conditions. The signal for atomic emission flame spectrometry is given by equation [47]. In seeking the optimum o X II o Pf E-4 .4 ,-4 .4 *I Hr ^0 I I- x I . \ 0 i-4 I x O 1 < 01 /- / x .4 01 0 i-4 x .4 *4 -I / / x / ^ 4-J n) oo II II c a 0 o \4 O o o oo 0 0o 0 a & O C) II a2. 0. SI I I I O S-4 <0-- --I i- - --I o 4-1 0 E-4 s-I s-I el 0 (a iCl -J cr m U) U) H -4 0 a o En 0 r. .re t0 ,-- 0U U r-i oc' CO '4- 0 r4 ,4 U -4U2 C 04 CO 4J 5-1 flame conditions the instrumental parameters, except W, will remain constant because the object sought is the flame type which will give the largest signal-to-noise ratio for a given instrument. Therefore equation [47] may be rewritten, using the optimum slit width expression, equation [53], to give i = kg 1c)-1/2I [76] where k TfH(A/F2)n [2BMecid 1/4 8 T(ITfH(A/F2) (af)1/2Rd)2J In the evaluation of k8 it is assumed that 1 is unity. The relative signal for each flame type can then be calculated by substitution of the proper value of lc and the proper integrated in- tensity expression into equation [761. However, the evaluation of I, the integrated intensity, is complicated by the fact that a plot of I versus sodium concentration would show two distinct regions of the curve. (See Appendix G for a further discussion of the intensity of spectral lines.) In the low concentration region the integrated intensity is proportional to the first power of N and is in fact given by hVogu I = o4 NAtLe-Eu/kT [77] 1074B (T) where all the terms have been previously defined. In the high con- centration (or self-absorption) region, the integrated intensity in watts/cm.2 ster. is proportional to the square root of N and is given by hvo2 -E /kT( LQDgu 1/2 I = 7 e u n aNAtL [78 o 107 -I-In 2 B(T) ' There is, of course, a small intermediate region in which the integrated intensity expression corresponds to neither of the above expressions. However, N can be converted to integrated intensity values with sufficient accuracy by extrapolating from the extremes to find the point of intersection of the two regions (i.e., some value of N which would give the same value of I by either expression), and using equation [77] for N values below this point and equation [78] for N values above this point. If equations [77] and [78] are equated at N equals Ni, then the intersection point, Ni, is found to be given by 16tAVD B(T) Ni n 2 AtLXogu [79] Therefore, for N values less than Ni the detector output signal for atomic emission flame spectrometry, i, can be calculated by the expression i = k8k9Ic)-l/2LNeu/kT, [80] i.e., equation [77] is introduced into equation [76], and hQogu k9 = 1074tB(T) At For N values greater than N. the detector output signal can be calculated by the expression i = kgklO(Ic)-1/2L1/2N1/2e-EukT [81] i.e., equation [78] is introduced into equation [76], and h90o2 ( ADgu ) 1/2 k10 = 107c n B(T) aA) cc c 'ln B (T) For convenience in further calculations, the constant kl0 can be written in terms of the constant k9. Of course, continuity exists at N = Ni, and so equation [80] and equation [81] will give the same value of i at the atomic concentration Ni, i.e., k8k9Ic)-/2LNieEu/kT= -k8kl0 c)-1/2L1/2Nil/m2eEu/kT [82] and solving for kl0 gives 1/2 1/2 kl0 = k9L /Nil [83 Substituting from equation [83] for kl0, equation [81] can be re- written to give S -1/2 1/2 1/2 -Eu/kT i = k8kg(-Ic) LNil 2- [84] Equations [80] and [84] are altered slightly if more than one spectral line of the species of interest lies within the spectral slit width of the monochromator and, therefore, reaches the detector. This situation arises in the case of the sodium resonance doublet (5890, 5896 A.), which is not resolved by the Beckman DU monochromator. The intensity of each of the lines making up the double is still given by equation [77] for the low concentration case and by equation [78] for the high concentration case. If 1I is the intensity of line 1 (5890 A.), and gl is the statistical weight of the upper state of line 1, then for low con- centrations hQ0g1 -Eu/kT [85 I1 =1074iB(T) NAtLe- If 12 is the intensity of line 2 (5896 A.), and g2 is the statistical weight of the upper state of line 2, then for low concentrations hO082 -Eu/kT 12 hVg2 NALeu/kT [86] 1074tB (T) Because Vo, At, and E are essentially the same for both lines, the same values can be used in both equations [851 and [86] without introducing significant error. For the high concentration case the intensities of lines 1 and 2 are given by hVo 2 -Eu/kT 1Dgl 11/2 hI 2 aN)t [87] S107c Eu/T ln B(T) t hVo2 eEu/kT AD2 1 /2 12 = n07c ( n B(T) aN [88] Equation [76] for the detector output signal for the case of an unresolved doublet is then given by i = k8GIc) -1/2 ( I1 22) [89] where V1 is the slit function factor for line 1 and K2 is the slit function factor for line 2. The evaluation of k for each spectral component can be carried out by use of equation [6] in Section II, i.e., = 1 i - ol [6] s For the Na 5890, 5896 A. doublet, when using the Beckman DU mono- chromator (Rd = 330 mp/cm. at 5890 A.), if the wavelength setting of the monochromator is 5892 A. (nearer the stronger line), and if the monochromator slit width is 0.01 cm., then s = RdW = 3.3 mp, Y1 0.94, and 2 = 0.88. Substituting the proper intensity expressions into equation [89] and collecting terms shows that for low concentrations the detector output signal for atomic emission flame spectrometry is given by h^ A i = k8 ) -1/2 hOAtT NLe-Eu/kT( g + 2g2) [90] i= 80,1c) 1074cB (T) eigl1+1 K292) 90] Collecting constant terms gives i = kgkll( c)-1/2NLe-Eu/kT, [91]. where hO A ot kll = 10743B(T) (lgl + 2g2) At high concentrations the detector output signal is given by h0o2 eEuT D aNAtL 1 /2 12 i = k__^1/E12 ) 1 + k2 (g2) ] 1 [92] Collecting constant terms gives i = k8kl2 c -1/2 1/2 1/2 -E/kT [ 93] where h 2 D A A \1/2 /2 o t [ (g 1/ kl12 107c \ n 2 B(T) [ 1 2l +2 By equating equation [91] to equation [93] when N equals Ni, it is possible to evaluate k12 in terms of kll, i.e., k2 = k lNil2L1/2 [94] and i is then given by i = k8kll -c)-1/2Nil/2N1/2Le-EukT [95] Equation [79] for the value of N at the intersection point, Ni, is also altered for the case in which the doublet is not resolved. In this case'Ni is given by 16B (T)AVD a[l (gl) 1/2 + 2(g2) 1/2 2 N =[96 i AtLXo2 V1-2 (~8gl + Vg2)2 It should be noted at this point that the optimum slit width for atomic emission flame spectrometry was derived with the assumption that the spectral line of interest was single, sharp, and isolated. This assumption is not valid for the analysis of Na using the Beckman DU monochromator. The value of I in equation [51] should be replaced by I = ii + V212 Because 1i and 12 are dependent on the slit width, W, I is in this case dependent on the slit width. Fortunately the dependence of I on W is slight because 11 and L2 are close to unity (see example of calculation of i and (2 above), and the dependence of I on W will always be slight when the spectral lines are close together, and the spectral slit width, s, is large. Calculation of the total root-mean-square photodetector output noise, AiT, for atomic emission flame spectrometry can readily be accomplished by assuming that Ic ( and the fluctuation in Ic, AIc) is approximately constant over the temperature range of any given flame. This assumption should not affect the results of the prediction of the optimum flame type, because the variation of I over the temperature range of a given flame is quite small, whereas the variation of I c in going from one type flame to another is quite great (17). When the proper optimum slit width is used in equation [48] for calculating AiT, the value of AiT will be approximately the same for each flame type. Qualitatively this could have been anticipated because optimization of the slit width results in obtaining the slit width at which the noise due to flame flicker (YTfWH(A/F2)AIc(Af)1/2s) is reduced to approximately the same value as the noise due to the dark current 1/2 shot effect ([2BMecid] 1). A decrease in the slit width below the optimum value does not result in an appreciable decrease in the noise, because the dark current noise does not depend on W. Because for a given detector the dark current is the same no matter which flame is used, AiT is approximately the same for all flame types if the optimum slit width is used. Table 6 shows values of W1 and AiT for three flames, calculated using the values listed under the table, which are typical values for a Beckman DU monochromator at 5890 A., a 1P28 photomultiplier, and a total consumption atomizer-burner. Because AiT is approximately the same for all three flame types, the flame type giving the maximum signal will also give the maximum signal-to-noise ratio. For low concentrations where the signal, i, is given by equation [80], the signal-to-noise ratio is given by i/Ai = kclIc)-/2LNeEuT [97] TABLE 6 OPTIMUM SLIT WIDTH AND TOTAL ROOT-MEAN-SQUARE NOISE FOR THREE FLAMES Flame Wo (cm.) AiT (amps.) H2/air 6.1 x 10-2 3.2 x 10-10 H2/02 1.7 x 10-2 3.1 x 10-10 C2H2/02 6.5 x 10-3 3.2 x 10-10 The following values were taken as typical for a Beckman DU monochromator and a 1P28 photomultipl er: B = 1.5, M = 106, ec = 1.6 x 10-19 coulombs, i = 10-7 amps., = 10' amps./watt, Tf = 0.5, H = 1 cm., A = 25 cm.2 F = 2500 cm.2, Rd = 330 mp/cm., Af = 1, AlI = 0.005 I, I = 7.7 x 10-10 watts/cm.2 ster. mp for H2/air (17), Ic = 9.7 x 10 watts cm.2 ster. mp for H2/02 (17), and Ic = 6.8 x 10-8 watts/cm.2 ster. mp for C2H2/02 (17). where kc = k8k9/LiT. For high concentrations where the signal, i, is given by equation [841, the signal-to-noise ratio is given by i/iT = kc c)-1/2LNil/2Nl/2e-Eu/kT 9[8] For the case of the unresolved doublet spectral line (equations [91] and [95]), at low concentrations the signal-to-noise ratio is given by i/AiT = kd(--c)1/2NLe-Eu/k [99 where kd = k8kll/iT, and at high concentrations the signal-to-noise ratio is given by i/ai = kd ) 1/2i/2N1/2Le-EukT [100] Figure 2 shows plots of i/aiTkc versus T. The upper dashed line in Figure 2 is i/ZiTkc at N = Ni, where Ni has been calculated from equation [79]. Below this dashed line k/AiTkc is calculated from equation [97]. Above this dashed line i/i-Tkc is calculated from equation [98. At low concentrations i/AiTkc = i/i-Tkd (Lc)- /2LNeEu/kT, i.e., the curves given in Figure 2 are valid whether a single spectral line or a multiple passes through the spectral slit width. In this low concentration region the plotted signal-to-noise ratios are entirely independent of the instrument used, i.e., the optimum flame type can be determined without knowing what instrumental setup is to be used. The absolute value of the signal-to-noise ratio for a particular instrumental setup, flame type, and flame temperature can be calculated by evaluating kc (or kd) by substituting values for the experimental parameters. a' 0 -I 0 '- xl -zj I 0 1-4 -4 0 -4 xl CO 'I C- SI I I r 0 0 c-1 u 0 <'- -4l 0 J0 I-4 -o *4 3 00 1-I oo -1 o0 Zr-e 0 L U0 inm t 0 " EW ,-4 1 o .U -40 4 C/ 0 C'I 4-1 cfE At high concentrations the plotted signal-to-noise ratio depends to a small extent on the instrumental setup because i/AiTkc depends on Ni, the value of which depends on whether the doublet is resolved or unresolved. When the spectral line is single (as the case would be when using the 0.5 meter Ebert mounting, Jarrell-Ash monochromator), Ni is calculated from equation 179]. When the spectral line is an un- resolved doublet, Ni is calculated from equation [96]. The difference in values of Ni as calculated from equations [79] and [96] is small enough that neglecting the instrumental dependence of Ni will not affect the choice of the optimum flame type. This is especially true because the signal-to-noise ratios i/AiTkc or i/AiTkd depend only on the one- half power of Ni. Neglecting the small instrumental dependence of the signal-to- noise ratios plotted in Figure 2, much information about optimum con- ditions for analysis of Na by atomic emission flame spectrometry may be obtained from Figure 2. The two dashed lines for each flame mark the upper and lower limits of an optimum region for that flame. The upper dashed line (S.A.), as previously discussed, indicates the beginning of the self-absorption region. Above this line the signal, and hence the signal-to-noise ratio, is proportional to the square-root of N, the atomic concentration. The lower dashed line (L. of D.) marks the limit of detectability for the Na 5890, 5896 A. doublet for the particular flame when using a Beckman DU monochromator and a good 1P28 photo- multiplier. The limit of detectability lines were obtained by calcu- lating the limits of detectability in atoms/cm.3 for Na in each of the three flames, at several temperatures, from equation [16]. From these values of N, i/i-Tkc values were calculated using equation [97]. The limit of detectability lines are the only lines on Figure 2 which depend to a great extent on the instrumental setup. The limit of detectability will be different with each monochromator-detector combination, and therefore this line must be calculated for specific cases. In the region between the self-absorption line (S.A.) and the limit of detectability line (L. of D.), the spectral line is detectable and self-absorption of radiation is negligible. In the H2/02 flame at 25000K. the point at which self-absorption becomes important corresponds to a PT value of approximately 4.4 x 10-6 atmospheres. At an aqueous solution flow rate of approximately 2 cm.3/min. and gas flow rates of 2860 cm.3/min. 02 and 10,000 cm.3/min. H2, this corresponds to a solution concentration of approximately 2.3 x 10-3 M, i.e., 53 p.p.m. (n298/nT = 1.2, = 0.5). Also, in the H2/02 flame at a temperature of 25000K., linear analytical curves for Na, free from the effects of self- absorption and ionization should be obtained for PT values from approximately 10-11 to 10-6 atmospheres (or solution concentrations of approximately 5.2 x 10-9 to 5.2 x 10-4 moles/liter). A H2/02 flame at a temperature of 25000K. should also give linear analytical curves for Na but with half the slope of the previous curves due to self-absorption for PT values above approximately 4 x 10-6 atmospheres. Optimum Flame Conditions for Atomic Absorption Flame Spectrometry The optimum conditions for analysis by atomic absorption flame spectrometry can be determined in much the same way as for atomic emission flame spectrometry. The signal-to-noise ratio for atomic absorption has'been given in equation [54]. After dividing both numerator and denominator by rTfWH(A/F2)I, equation [54] can be rewritten as (io it)/ZT = 1 e-pkOL S T2BMec 1i /2 Af o + (0fo/Io)2 + (Zgcs/IO)2 + (sL/Io)2 [1101] arWH(A/F2s [loll When the optimum slit width, Wo, is used in equation [101], (-I-cs/Io)2 is much smaller than (7o/I)2 and can be neglected. Sub- stitution of values in the first term of the sum in the denominator of equation [101], shows that for any practical set of conditions this term will always be quite small when compared to (A70/Io)2, which will never have a value smaller than about 10 4. The last term, (Is/Io)2, will be small in most cases of practical importance in atomic absorption flame spectrometry. The intensity of the scattered radiation Is is always a fraction of I1. Usually the fraction is quite small, and the scattering term can be neglected. The fraction of scattered radiation could always be made small by using organic solvents or chamber-type atomizers to decrease the particle size of the aspirated solution in the flame. For the purposes of selecting the optimum flame conditions, equation [101] can be rewritten as (io it)/T = k3(1 e-PkoL) [102] where k13 = [(o/i)(f)1/2] -1 The value of ko, the atomic absorption coefficient at the line center (in cm.-1), is given (31) by 2 2 n 2 X 02gNAt ko = [103 ] a0D T 8t B(T) An expression for the Doppler half-width, AVD, has already been given in Section III: S2 2R In 2 T )l/2 D= --c o0 45] After substituting AL~ from equation [45] into equation [103], and the resulting expression for ko into equation [102], (io it)/AiT is found to be given by (i it)/AT = k13(l e-k4NL/T/2) [104] where the terms which are not dependent on flame temperature or flame composition have been collected into the constant k14, and p2 l- 2 0o2guAticMaI/2 k4 = ,- 8rB(T) 2 v2R In 2 V 0 If the spectral line of interest is not a singlet but a doublet, as in the case of the Na 5890, 5896 A. double, which is not resolved by many monochromators, then equations [102], [103], and [104] must be altered. Equation [102] should read in this case (io it)/-T = k13(l e- k0L) [105] where e is a factor to account for the line being multiple. The evaluation of e is discussed in Appendix H. The value of I in the constant k13 is the sum of the intensities of the two lines making up the doublet, i.e., 10 = l1IOl + 2I02, where line 1 is the 5890 A. line, and line 2 is the 5896 A. line. The value of ko is also altered. For a doublet ko = kol + k02, where kl0 and k02 are the atomic absorption coefficients at the line centers of line 1 and line 2, respectively. Therefore, equation [103] should read 2 ~1n 2 x02 NAtj ko= A~D 3-T8 B(T) (gl 8 82) [106] where gl and g2 are the statistical weights of the upper states of lines 1 and 2, respectively, and 0o, At, and A9D are approximately the same for both lines. Equation [104] should read in this case (io it)/T = k13(l e-kl5NL/T/2) [107 ] where e 2 1n 2 02(gl g2)AtS cMal/2 k15 = -k1 8jr B(T)2 J2R In 2 It may be seen from equations [104] and [107] that no matter which of the two expressions most accurately gives the value of the signal-to- noise ratio for atomic absorption flame spectrometry, when flame con- ditions are such that NL/T1/2 has its maximum value, the value of (io it)ZT will also be at its maximum. Figure 3 shows plots of log (NL/T1/2) versus T for three flames and seven values of PT. These plots have the advantage of being entirely independent of the instru- ment used, i.e., the same set of curves can be used to determine optimum flame conditions no matter what spectrometer is to be used in the oro r II I I 0 0 0C * x x H- I o P4 x IaI [-4 x p.' o00 I 0 -4 'zP I I I I - * I I xI x -4 1 Ij -j 4-) d E . C, ) I d 0 -4 C M Ci 0 T -W 0 - N U c IOn CO X E -I SII - ci 0 N O- 0 " 00 ------ 0 E--l M l-1 l O 0 C) 0 en o o o co N O 0 O 4-W C) Sli o o 0 0 co N 00 Cd O 0 r-4 o 0 0 N 0 0 0 0 0 o -4d 0 0 -4 O^ 0 .-4 C) co c4 S-1 0 E-H 0 r-4 0 Cd 0 0 4d -4 ) E-4 z 0 0) 0 ca a) Cd i- ed O 0 I analysis. The absolute value of the signal-to-noise ratio for atomic absorption can be obtained from Figure 3 by evaluating the constants in equations [104] or [107]. The dashed lines in Figure 3 mark the limits of detectability for the Na 5890, 5896 A. double for the particular flame when using a Beckman DU monochromator and a 1P28 photomultiplier detector. The limits of detectability were calculated from equation [44]. The limits of detectability are, of course, dependent on the instrumental setup used and must be calculated for specific cases. Although it is not indicated in Figure 3, there is an upper limit to the optimum region for analysis by atomic absorption flame spectrome- try. Qualitatively it may be seen that an upper limit would be reached when the atomic concentration, N, reached a value such that the difference between the output signal due to the transmitted intensity when solvent is introduced into the flame, 10, and the output signal due to the transmitted intensity when sample is introduced, It, becomes of the same order of magnitude as the noise. An equation for the maximum detectable concentration can be derived in much the same manner as for the minimum detectable concentration. However, the resulting equation is unduly complex and can only be solved graphically. It is not particularly important to be able to calculate the maximum detectable concentration because it is always possible to dilute concentrated solutions in order to work in a more favorable concentration region, and so the theory for the maximum detectable concentration is not presented in this paper. Calculations The equilibrium constant for the dissociation of NaC1, K1, as a function of temperature, is calculated from the expression given by Mavrodineanu and Boiteux (30), log K = -5040 DNaCl + log T + log (1 10-0625e /T) T 2 + iNa iC l iNaC1 [108] where DNaC1 is the dissociation energy of NaC1 in electron volts, WOe is the vibrational constant of the molecule in cm.-I (21), and iNa' iC1, and iNaCl are the chemical constants of Na, Cl, and NaC1, respec- tively. For a monatomic gas (either Na or Cl) the chemical constant is given (30) by iM = -1.587 + i log Ma + log go [109] where Ma is the atomic mass of the species of interest, and go is the statistical weight of the ground state of the atom. For a diatomic gas such as NaC1 the chemical constant is given (30) by iNaC2 = -1.738 + -log NaC1 + log Mg log Be,[ll0] where MNaCl is the molecular weight of NaC1, go* is the statistical weight of the ground state of the molecule, and Be is the rotational constant in cm.-1 of the molecule (21). The equilibrium constant for the ionization reaction, K2, as a function of temperature is also calculated from an expression given by Mavrodineanu and Boiteux (30), log K2 = -5040 V/T + 5/2 log T + log go' log go 6.1818, [111] where V is the ionization energy of the atom in electron volts, go is the statistical weight of the ground state of the neutral atom, and go' is the statistical weight of the ground state of the ion. The equilibrium constant for the dissociation of NaOH, K3, as a function of temperature is obtained from thermochemical data tabulated in the JANAF Tables (25). The values found for KI, K2, and K3 are summarized in Table 7. The partial pressure of flame electrons, pe, is taken to be 4 x 10-9 atm. (15) in the outer cones of all three flames. The partial pressures of OH in the various flames are estimated from the information given by Zaer (46), and pOH is assumed constant over the temperature range of the individual flames. In estimating the values of POH, it is necessary to take into account the solvent being aspirated into the flame because the solvent contributes a large portion of the flame gases. For the C2H2/02 flame, POH is estimated to be 0.2 atm. In the H2/02 and H2/air flames POH is estimated to be, respectively, 0.05 atm. and 0.001 atm. The value of TIc is estimated to be 5 x 10-31c, where Ic has been measured for the various flames by Gilbert (17). Atomization efficiencies and flame temperatures are estimated from work performed in this labora- tory (42,43). In calculating Figures 2 and 3, L is taken to be 1 cm. in all cases, because this is found to be approximately the radius of a flame in good adjustment. The value of Eu for Na is 2.1 electron volts (28). TABLE 7 VALUES OF K1, K2, AND K3 AS A FUNCTION OF T T (OK.) K1 (atm.) K2 (atm.) K3 (atm.) 1600 4.4 x 10-7 2.5 x 10-15 1.3 x 10-7 2000 9.1 x 10-5 6.4 x 10-12 4.1 x 10-5 2400 3.2 x 10-3 1.5 x 10-9 2.0 x 10-3 2800 4.2 x 10-2 7.4 x 10-8 3.1 x 10-2 3200 1.4 x 10-1 1.6 x 10-6 2.5 x 10-1 In calculating Ni from equations [79] and [96], the value of a is taken from data given by Hinnov and Kohn (22). The Doppler half- width, A D, is calculated from equation [45], and B(T) is equal to go, the statistical weight of the ground state of Na. Spectral data, such as At, gu, and go, are taken from the Landolt-Bornstein tables (28). Experimental Verification of Theory The validity of the theory developed in this paper has been shown by comparing theoretically predicted analytical (working) curves with experimentally determined analytical curves. The theoretical analytical curves were obtained from Figure 2 by choosing a temperature typical of the flame to be used (43) and then reading off PT and corresponding lof (i/AiTkc) values at that flame temperature. Because the signal-to- noise ratio is proportional to the signal, as previously discussed, plots of log (i/AiTkc) versus log PT will have the same shape and slope as the analytical curves of signal versus concentration. The experimental curves were obtained by recording the output signal for a wide range of solution concentrations. Solution concen- trations were converted to Pt values by equation [581 so that measured values of signal as a function of Pt were obtained. The measured curves of the logarithm of the signal versus log PT were shifted along the log (i/AiTk ) axis until the experimental curve for the C2H2/02 flame coincided with the theoretical curve. The analytical curves for the H2/02 and H2/air flames were shifted by the same factor as the C2H2/02 flame curve. Agreement as to shape and slope of the two sets of curves for all three flames is quite good as shown in Figure 4. S0 c"! --*o-I o *o ,, om I II 1 0 C) o c 00 0 I 6 0 0 f O0 U r-0' II C 0 -Ii cu 0 0 a' C4 OO N ,- II 0. CO 1N II C O CM Lj CN CN o II II o x0 -o.. .- 0 o 5l a 0 H co ,o - o Yo o Q C z c LO o E o \ I --' * o, *. o " S*. O C * SO cI u II rz co c ou c So 0 o0 ' 41 \ n r s 4 ` c, c O I I h ,-O ** oC SOrII C ) g O < 0N O II II 0 * 0 d *-o- o a 0 I * 0 rI OC S1 r0C I \ u SN-. o u 3 CY0 7 uH Su0 *4 .u 0 o o- 0 cc 0 II * C- II -- 0 CM r0 CM *I- 0 II I -4 I 4 :; a )H r- It should be noted that Alkemade (1) predicted the shape of the sodium 5890, 5896 A. flame emission analytical curve by deriving the "curves of growth" for the total absorption of the Na doublet. He had good correlation between the derived curves and the experimental curves. Alkemade (1), however, neglected the effects of ionization and compound formation on the shapes of the analytical curves and also did not cal- culate the intersection point of the curves. He used experimental conditions substantially different from those used in this paper; namely, a propane/air flame, a chamber type atomizer-burner, and a glass trans- mission filter for isolation of the sodium doublet. Even though experi- mental conditions were considerably different from those used in this paper, the basic theoretical results were quite similar. The experimental analytical curves were determined on a Beckman DU with spectral energy recording attachment (SERA) and 1P28 photo- multiplier detector. Efforts were made to have the measurement pro- cedures agree as closely as possible with standard practices in order to illustrate the practical value of the theory developed in this paper. Analytical curves were determined for four flames with conditions as noted on the curves. The analytical curves shown in Figure 4 are average curves for quadruplicate measurements. The Beckman flame housing attachment was used throughout, but the mirror in the housing was blocked so that only radiation coming directly from the flame entered the spectrometric system. Blocking of the mirror was necessary to insure reproducible entrance optics. Over the wide concentration range used in this study, the relative contributions of radiation arriving directly from the flame and radiation reflected by the mirror changed as the concentration changed because of the variation of the amount of reflected radiation absorbed in passing through the flame gases. It was found that the ratio of meter reading with mirror unmasked to meter reading with mirror masked varied from 3.18 at 10-1 p.p.m. Na to 2.06 at 1000 p.p.m. Na. If the mirror were not blocked, this would lead to curvature of the analytical curves not accounted for in the calculation of the theoretical curves. However, the influence of entrance optics could be accounted for by supplying the proper value of n, the entrance optics factor, to adjust the measured signal values (see equation [47]). In determining the experimental curves the calculated optimum slit widths were used, and the sensitivity control was adjusted to keep the readings on scale. With the Beckman housing a large region was observed by the monochromator (approximately 1.6 cm. wide by 2.6 cm. high). With the setup used, the lower edge of this region was 1.1 cm. above the tip of the burner. The temperatures indicated for the differ- ent flames were estimated as average temperatures over the flame region viewed (43). The solutions used in constructing the experimental curves had -2 concentrations of 10-2, 10 -, 1, 0, 100, and 1000 p.p.m. The solution of 1000 p.p.m. Na was prepared by dissolving the appropriate weight of NaC1 in distilled water. The other solutions were prepared by successive dilutions. Lower solution concentrations were not used because of difficulties encountered with spurious Na emission, apparently attributable to dust particles in the air. As may be seen in Figure 4, the position of the intersection points and the slopes of the experimental and theoretical curves agree quite well. Because the H2/02 flame with the lower flow rate of H2 has a radius approximately half that of the other flames, the theoretical intersection point for this case was calculated using L = 0.5 cm. This flame does not represent a practical case because the size of the flame is too small to be experimentally useful. However, it is interest- ing to note that the theory quite adequately accounts for the effect of flame size on the point at which self-absorption becomes important. Note that in Figure 2, only one value of pOH is given for H2/02 flames. Measurement of the OH emission in the outer cone of the flame at 3090 A. for both H2/02 flames shown in Figure 4 indicated that the value of POH was approximately the same in both cases. It should not be too surprising that pOH is the same in the outer cones of both flames, because there must be considerable entrainment of air in the turbulent flames produced over total consumption atomizer-burners. The OH in- tensity measurements were performed with the flame masked by a flat black baffle with an opening of approximately 2 mm. by 3 mm. The meter reading for the "fuel-rich" flame was approximately twice that of the "stoichiometric" flame, as would be expected for two flames at the same temperature, with the same POH, and a radius ratio of 2 to 1. Figure 4 also shows that the theory is quite adequate in predict- ing the signal-to-noise ratios for the various flames. All the experi- mental points were shifted along the log (i/AiTkc) axis by the same factor, and the experimental points for each of the flames agree fairly well with their respective theoretical curves. Therefore the measured signal-to-noise ratios of the various flames must be related to each other in th- same way as the theoretical signal-to-noise ratios. For example, for log PT = -6, the theoretical values of log (i/AiTkc) for H2/air, H2/02 (curve 1), and C2H2/02 are 12.3, 12.5, and 12.0, re- spectively, while the experimental values are 12.1, 12.6, and 11.9, respectively, i.e., the theoretical values of the three flames are in the ratio of 1.02/1.04/1.00, while the experimental values are in the ratio of 1.02/1.06/1.00, approximately the same, within experimental error. The theoretical curve for the "stoichiometric" H2/02 flame (curve 2) has been adjusted to account for the size difference. Because the region viewed by the spectrometric system is wider than the flame width, the signal should increase as the square of the flame radius in going from the "stoichiometric" to the "fuel-rich" flame. The "fuel-rich" H2/02 flame has a radius of 1 cm., the "stoichiometric" H2/02 flame has a radius of 0.5 cm., and, therefore, the size correction should be a factor of 4. There is also a small temperature difference between the two H2/02 flames. In the smaller flame the temperature must be averaged over almost the entire flame, while in the larger flame the temperature must be averaged over a smaller portion of the flame. (The smaller flame was approximately 4 cm. long, while the larger flame was approxi- mately 8 cm. long.) The excellent agreement of the theoretical and experimental analytical curves also demonstrates that the influence of species such as NaO and NaH must be negligibly small. The agreement of theory and experiment also indicates that, contrary to many statements to be found in the literature, the formation of the NaOH species in the flame gases is not negligible. The essential correctness of the equation for optimum slit width for atomic emission flame spectrometry, equation [53], has been verified by measuring the noise as a function of slit width. The peak-to-peak noise was measured at several slit widths above and below the calcu- lated optimum slit width. The peak-to-peak noise was multiplied by 1/2 42 to obtain root-mean-square noise, and the ratio of W to root- mean-square noise was plotted versus W. For a constant intensity source, the signal should be proportional to W (see equation [47]), and W/noise should be proportional to signal/noise. In Figure 5 the unbroken line indicates the calculated variation of log (W/AiT) as a function of W for a H2/02 flame. The value of W/AiT is given by W/ai = W [2BMef(id + YTfWH(A/F2)Ics) + ( TfWH(A/F2)Afcs)2]f ] -1/2 [112] The circles in Figure 5 indicate measured values of log (W/noise) for a H2/02 flame. The measured values of W/noise were shifted along the signal-to-noise axis to obtain overlap of the experimental and theoretical values in order to compare curve shapes. Agreement of theoretical and experimental values is adequate. The scattering of experimental points is due to the difficulty of measuring accurately the small values of noise. All noise measurements were made at 5800 A. The H2/air and C2H2/02 flames gave signal-to-noise ratio versus W plots similar to the H2/02 flame plot. A Moseley x-y recorder (Model 135, F. L. Moseley Co., Pasadena, California) was used for all noise measurements. S5.6- I 0 5.2_ a 4.8- S4.4 S4.01 - 5 1.2 1.4 1.6 1.8 2.0 W (cm. x 102) Figure 5. Calculated and Experimental Plots of Signal-to-Noise Ratio Versus Slit Width for Sodium in a Stoichiometric H2/02 Flame. Because of the instrumental problems inherent in atomic absorption measurements of the alkali metals and because of the greater appli- cability of atomic emission flame spectrometry for the analysis of sodium, no experimental measurements of absorbance as a function of concentration (or PT) were made. However, because the theoretical atomic emission analytical curves agreed quite well with the experimental curves, it is highly probable that'the theoretical and experimental atomic absorption analytical curves should also agree very well. The emission analytical curves depend directly upon N, as do the atomic absorption curves, and so the verification of the emission curves indirectly verifies the absorption curves. V. CONCLUSIONS It is hoped that the-work presented in this dissertation adequately demonstrates that both atomic emission and atomic absorption flame spectrometry can be treated quantitatively in a relatively simple manner. The equations discussed allow one to predict the effects of variation of instrumental slit width, the effects of compound formation and ionization in the flame, and the effects of self-absorption of radiation. This quantitative treatment makes it possible to predict the shape of analytical curves and, therefore, to choose conditions so as to obtain linear analytical curves. It is also possible to calculate values of limits of detectability, signal, noise, and signal-to-noise ratios for any instrumental setup. Information on optimum experimental conditions for atomic emission and atomic absorption flame spectrometric analysis of any spectral line of any atom of any compound introduced into any flame and measured using any experimental setup could be presented in the form of graphs such as those in Figures 2 and 3. With accompanying data on optimum slit widths and limits of detectability for particular instruments, this information should be extremely valuable to many analysts using flame spectrometry who are not necessarily specialists in the field. Enough data are presently available that a treatment such as has been presented in this paper for sodium could be applied to many elements of interest in flame spectrometry. Preliminary calculations have already been carried out on Li, K, and Mg. The results of these calculations will be tested experimentally and presented at a later time. The quantitative approach presented in this paper certainly would seem to be a faster and surer method of obtaining optimum conditions than the trial-and-error methods presently employed. It should be noted that the essential agreement with experiment of the theory presented in this paper should be good indirect evidence for the existence of chemical and thermal equilibrium in the flame, because equilibrium was assumed in the development of the theory. The theory of atomic emission flame spectrometry presented in this paper could also be extended to situations in which the excitation of radiation is non-thermal, e.g., excitation by chemiluminescence, by substitution of the proper expression for I in equation [76]. In fact, the general approach discussed in this paper could be extended, with good purpose, to many other areas of spectrometry. The case of the d.c. arc source is only one example of an area which could benefit greatly by being subjected to the quantitative approach presented in this dissertation. APPENDICES APPENDIX A Derivation of Current-Intensity Expression The output signal of a photodetector, i.e., anodic current, in amperes, as a result of monochromatic radiation (spectral line half- width is assumed to be much narrower than the spectral slit width, s) incident on the entrance slit of the monochromator, can be derived by consideration of entrance optics, monochromator optics, and the detector sensitivity. If the total (integrated) intensity of a spectral line is denoted as I and has units of watts/cm.2 of source steradian, then the total power of radiation reaching the monochromator entrance slit as a result of this spectral line is given by entrance I (area of source).(solid angle viewed by monochromator). [113] If the monochromator entrance slit is fully and uniformly illuminated by the radiation from the actual source, then the slit acts as the effective source, and so the area of the source is given by the area of the slit. The solid angle viewed by the monochromator is given by A, the effective aperture of the monochromator, divided by F2, the square of the focal length of the collimator, as long as the effective aperture is fully illuminated. Because of reflection and absorption losses within the mono- chromator, only a fraction Tf of the radiant power reaches the exit slit. If the exit slit is equal to or slightly larger than the entrance slit, then the total power of radiation at the exit slit is Pexit = Pentrance Tf watts [114] Using the detector sensitivity factor r in amperes output at the anode per watt of radiation incident on the photocathode, the output current, i, from the phototube can be found, i.e., i = P exit amperes [115] exit The output current of a photodetector in amperes as a result of polychromatic or continuous rather than monochromatic radiation in- cident on the monochromator entrance slit can be derived in a manner similar to the above case. However, in this particular case an addi- tional factor must be included, namely the spectral slit width, s, of the monochromator. This is a result of the intensity of a continuum, Ic, being expressed as watts/cm.2 of source ster. wavelength inter- val, and so in this case the total power of radiation reaching the monochromator exit slit as a result of a continuum being incident on the entrance slit is given by Pexit = Ic (area of source)-(solid angle viewed by monochromator). (spectral slit width) Tf watts [116] where the spectral slit width, s, is the wavelength interval passing through the exit slit for any particular wavelength setting. The current, ic, due to the continuum can then be found by introducing equation [116] into equation [115]. |