Calculatioin of the Limit of Detectability in Atomic Absorption Flame Spectrometry J. D. WINEFORDNER and T. J. VICKERS' Department o f Chemisiky, University of Florida, Gainesville, Fla.
An expression is derived in this paper relating the minimum detectable concentration in atoms/cc. of flame gases and in moles/liter of solution to the experimental conditions for atomic absorption flame spectrometry. An expression is also derived for determining the optiimum slit width, and sample calculations are carried out to illustrate the use of the derived expressions.
I
a n expression for the limit of detectability in atomic absorption flame spectrometry will be derived in a manner similar to that used by the authors (66) for the limit of detectability in atomic emission flame spectrometry. The development of a limit of detectability expression is of importance to both theoretical and experimental work in the field of atomic absorption flame spectronnetry. Such a n expression allows a direct calculation of the sensitivity of analysis, which may be expected, as well as the calculation of parameters such as transition probabilities if the limit of detectability is known. However, of equal importance to the analyst, it makes possible the estimation of the effect of various experimental 1)arameters on the sensitivity of analysis. The latter greatly aids in the selection of optimum ex1)erimental conditions for analysis. N THIS PAPER
THEORY ON THE LIMIT OF DETECTABILITY IN A T O M I C ABSORPTION FLAME SPECTROMETRY
The system under consideration in the development of this theory consists of a hollow cathode discharge tube (HCDT), or other resonance lamp which emits sufficient;!y narrow lines, a flame into which the sample is aspirated, a lens for focusing radiation from the source onto thie central part of the outer cone of the flame (which can be considered to be tqiproximately in thermal equilibrium), a second lens for focusing the transmitted radiation onto the monochromator entrance slit, a monochromator with accompanying optics and slits, a photodetector, aml~lifier,and readout (recorder, meter, etc.). The H C D T emits a line of 1 Present address, United States Army, Ft. lIcClellan, Ala.
intensity Io (in watts,'cm.* ster.), and the intensity of the transmitted radiation is I , (in the same units as 1'). The transmitted radiation is focused by the second lens onto the entrance slit of the monochromator so t h a t the slit is fully illuminated and the effective aperture of the spectrometric system is filled with radiation. I t is assumed that correction is made for thermally emitted and fluorescent radiation of the same frequency as the incident radiation so that I , is the true transmitted intensity-i .e. it does not include emission from the flame other than flame noise in the frequency interval, Affl over which the amplificaten1 responds. -11~0it 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, v, (the frequency of the absorption line peak in second-'), and that only radiation of the line of interest enters the spectrometric To simplify the use of the following equation, the units and definitions of all symbols are given in Appendis I. If the spectral line emitted by the source is much narrower than the absor1)tion line in the flame, then I" is related to I , hy Beer's law ~
where .A" is the absorbance, k" is the atomic absorption coefficient in cm.-l a t the line center for any atomic concentration, and I; is the length in cm. of flame gases through which the radiation is passed. Justification for the assumption of a narrow soui'ce line is to be found in the statenients of Jones and FTrakh ( I d ) and Crosswhite (6) and in the ability of the experimenter to adjust the source line width by rhanging the operating current of the tubc, or, if necessary, even by cooling the tube. However, see dlq)endis I1 for a further ion of the effect of line width. t is seldom necessary to operate under conditicns in which the effect of Doppler, Lorentz, and resonance line broadening ( 1 4 ) 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 Al)pendix
111) that the effect of hyperfine structure is to reduce the absorbance belox the expected value. Thus A"
=
log (1')'It)
=
0.43pk"ZJ
(2)
where p is the correction factor to account for the decrease in absorbance. The method of calculating p is indicated in hpi)endis 111. The absorbance will he said t o be just detectable when the difference hetueen the detector output i g n a l (anodic current of the photodetector) io, in amperes, due to I" and the detector output signal it, in ainperes, due to If equals twice the total root-meansquare noise fluctuation. I t i.q assumed in this theory that the noise is nieasured a t the wavelength of the line of interest with solvent nsi)irntcd into the flame. Thus the minimum detectable signal difference is given by -
(io - it),,, = 2aiT
(3)
where & is the tot :il root-mean-square noise fluctuation in the oiltijut current of the photodetector in aniljeres, and the subscri1)t V I intliwtes that the signal diffei,enc$eis that observed a t the minimum detectable concentration. The relation.qhip b e t w e n Fignal in amperes and intensity in watts, cm.2 ster. has been derived in the earlier paper by Wincfordnw anti Vickws (26). Thus it may be shown that io -
-/T!WH(.t 'F')I"
(4)
and
where y is the photodetector sensitivity factor-+.e.. the current in amperes I)roduceti a t tlie anode of the detertor for each watt of radiant liower incident upon the Iihotoctathotle-T, is the tranwii.sion factor of the sliect e n , 11; is the monochrodth in rni.--i.e.> entrance and esit ylit \r-itlth* a1'e as.suined the same (&6)--Ii is the slit height in ['in,, :I is the effective apertuw of the monochromator in cin.2, and 1.' is tlie focal length of the collimator in cni. The term A V'* i,s the number of viewed by the si)ec3tivnietric long ah the effecatii-e aljerture of the monochromator is fillcd with radiation, VOL. 36, NO. 10, SEPTEMBER 1964
1947
Substituting for ioand it in Equation 3 one obtains yT,!$'ff(--l/F2) ( I " - I ! ) , = 2 A i ~ (6) However, from Equation 2, I , can be written in t e r m of I " to obtain
yT,T.17H(A/P)Io(I - e - p k m o L ) = 2AiT
(7)
where km0 is the atomic absorption coefficient a t the line center a t the minimuiii detectable concentration. Exinnding the exponential term and noting that k,"L ip small a t the limit of detection-i.e., e - - p k m o r , 2 1 - pkmoLone may write (A'i /F')I"pk,"L
y
-
=
2Ai T
(8)
The root-mean-square noise signal, A i T I may be attributed to flnctuations originating in the photodetector, fluctuations in the source, Aio! and fluctuations in the flame continuum, Aic. In this theory, it is also assumed that the decrease in intensity due to scattering of radiation by particles in the flame is negligible [see .-lppendix IV). The noise signal5 add quadratically (g), anti therefore, A i T is given by .AiT = ( ~ i , 2 2 0 2 Tic2)1/2 (9) .-
sp,
+ + The value of sp is given (I?)
by
Aip =
where is the fluctuation in the intensity of the continuum emitted by the flame in watts ster. mp second-"'! and s i R the effective spectral slit width (in nip) of the The spectral slit width is given (26) by the expression
s
+ + ic)]l'z (10) where B is a constant approximately equal to I + L'G + 1 'G2 + . . . , where
G is the gain per stage of the photodetector, .If is the total amplification factor of the photodetector, e, is the electronic charge in coulombs, id is the photodetector dark current in amperes, and i, is the anodic current of the photodetector in amperes due to the continuous emission of the flame. I n any practical case in atomic absorption flame spectrometry i dand i, will be much smaller than io or. a t least. can be made much smaller a t the limit of detection. Hence one may write -
(11)
and substituting from E:quation 4 one obtains for Aip'
hip2= 2B.lfe,AfyTfTt'H (A, F 7 ) I " (12)
= (s,2
+
s,2)1;2
(15)
where s, is the spectral slit width in mp as determined by the mechanical slit width, and s, is the spectral slit width in nip a t 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 15 differs from the one given for s in the emission paper (26) since in the atomic absorption case, the more exact equation for s can be used without overly complicating the theory. For wide slits s is determined almost solely by s,, and in this case s is given by
s =
[2B.lfeeAf(id io
Aig = (2BMe,Afi"j1'2
signal and the response bandwidth, As, in second-', has been more fully discussed el5ea-here (9, 26). Similarly, GCis given (26) by Aic' = [ y T , W H (A 1F 2 )Z L s 2Af ] (14)
S,
=
W/DF
=
RdW
(16)
where D is the angular dispersion of the monochroniator in radians Imp and Rd is the reciprocal linear dispersion of the monochromator in niplcm. A more complete discussion of spectral slit width is to be found in the preceding paper (26). I t will he convenient in calculating the noise t e r m to write the fluctuation as fractions of the total intensity such that TIo = X I " and TIc = € I c , where and t are the appropriate fractions, and IC is the intensity of the continuum emitted by the flame in units of watts,' cm.2 ster. nip. Therefore, Equations 13 and 14 may be rewritten in the following form : -
Ai.'
=
[yT,T~'H(d;'F2)~I'12Af (17)
-
AiC2= [ y T IV H (.-I I F 2 )[ I C s]'Af
(18)
Substituting the ai)proi)riate4.alueF into Equation 9, one obtains for AiT
Azo2
=
[yT,TfTH(,4 P ) P ] ' A f
ANALYTICAL CHEMISTRY
i,
E , is the energy of state i above the ground state, and the summation is overall states of the atom. For the case of the minimum detectable concentration. .Ym0 = S , g , , ~ B ( T ) . As previously discussed in the paper on the detection limit in atomic emission flame spectrometry, 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 S,' is essentially the same as S,. The above theory is still correct for transitions arising in levels higher than the ground state if go is replaced by the statistical weight of the state involved and if the Boltzmann exponential factor ( e - E i / k T ) is included in the Ihltzmann equation. Substituting for k,' and xiTin Equation 8, and solving for X, one obtains
N,
=
53.54uoB( T ) pLXo2g,'4rs x2
(13)
mhere goi b the fluctuation in the intenyity of radiation from the iource in unit> of natt. cm Yter .econci-1'2 The pecond-' unit arise- hecauie the noi-e i b a root-niean-quare noiie The relation.hqi hetn een the noise
1948
i
g, is the statistical weight of the state
+ +
The value of A t o 2 i y related to the fluctuation in the intensity of incident radiation (26) hy -
where A V O is the Doppler half width (in second-') of the absorption line, A, is the wavelength (in cm.) of the line center, g. and go are the statistical weights of the upper> Z L , state and the lower (ground, 0 ) state, .-I ! is the transition probability in second-' for spontaneous emission-i.e.: state u to state -and 6 is a factor to account for line broadening other than Doppler broadening and hyperfine structure broadening (see Appendix 111). Because analytically important atomic absorption transitions usually involve the ground state, only transitions to that state will be considered. The constants have been evaluated in the right hand part of Equation 20. The number of ground state atoms per cc. of flame gases is related to the total number of atoms in all states per cc. of flame gases, S,by the famous Roltzniann equation (26),namely So= Sg, ' B ( T ) where go is the statistical weight of the lowest (ground) state and B ( T ) is the partition function of the where atom-i.e., B ( T ) = zg,e-E"/"T
The value of kmo,the atomic absorption coefficient a t the line renter a t the minimum detectable ground atate concentration &Y,,,'IS ) given ( 1 4 ) by
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
.\ppendix IV the influence of thermal t.niission, fluorescent emission, and light scattering are also considered. The o1)timutn slit width, HT0,can be obtained by minimizing Equation 21 with respect to ll7-i.e.) equating dS,,'dll' to zero and solving for TI'. If s is given by s = [(R,117)2 s,2]1/2fromEquations 15 and 16, and if s, is iassumed constant with change in slit width, then Wo, in em., is given by
+
The minimum (letel-table concentrations in atonis;cc. of flame gases (S,) can be converted to minimum detectable solution concentrations in molesiliter ( C , ) by use of the expression previously derived (26). Thus
c,
=
3.3 x 1 0 - 2 2 . Y m & T n r / @ ~ ~ g(23) ~~p
where Q is the flow rate of unburned gases in cc.,'second a t room temperature and a t one atmosphere pressure, T is the flame temperature in OK., n~ is the number of moles of cornbustion products ht temperature T , nz9xis the number of moles prehent a t 298' K., @ is the flow rate of solution in cc.,jminute, E is the atomization efficiency, ;and pis a fraction to account for inconqilete compound dissociation and atomic losses due to ionization. The constant contains the numerical factors 298'' K., Avogadro's number, and the conversion factors from seconds to minutes and from cc. to liters. I t thus has units of (seconds minute) (cc ./liter) / (atoms/mole) (degree). DISCUSSION A N D SAMPLE CALCULATIONS
When one attempts to calculate minimum detectable concentrations through the use of Equations 21, 22, and 23, thc need for further measurements of basic exl)erimental parameters of atomic absorption flame spectrometry becomes apparent. One of the most iml~ortantcontributions of these equations is that they point out those areas most in need of further work. I t is difficult to compare calculated limits of detection with measured limits of detection in terms of solution concentrations unless accurate absolute values of I " , I,, xI E , nT, nzg8, E , and p, and the photomultiplier tube characteristics are stated. Also, more information is needed on 6 values, and on the degree of compound dissociation and atomic ionization so that /3 may be calculated (26)with some accuracy. Although the calculation of limits of detection is a t present hampered by this lack of information, Equation 21 is still valuable in that it allows one to calculate the effect of varying one or more of the experimental parameters on the sen-
sitivity of analysis-e.g., the effect of slit width or temperature on S,. In Table I the results of calculations of the minimum detectable concentrations for two favorable cases are given. When one chooses a good imtrument such as the Jarrell-.ish 500-mm. grating monochromator, a 1P28 photomultiplier detector, and flame conditions such as stoichiometric oxyhydrogen or oxywidth, all the terms under the square root in Equation 21 except x 2 are negligible. I t is assumed that correction has been made for the output signal of the detector due to thermal and fluorescent erni.ssion and light scattering. Thus for these conditions, which are of experimental importance, the value of S, depends directly on x the fluctuation of the source intensity, and so Equation 21 can be reduced to
Equation 24 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, which is valid if the emission line of the source and the absorption line of the atom in concern are .single and spectrally isolated. This, of course) will not be true for absorption measurements on the Ka 5890to 5896-A. lines when using medium dispersion prism inhtruments such as the Beckman DU monochromator. If the absorption lines are not resolved, then it is necessary to account for the intensity of the source and the absorbance of the atomic gases as a function of monochromator setting, spectral slit width, s, and mechanical slit width, 11.. Therefore the data in Table I have been obtained assuming a grating monochromator, such as the Jarrell-.ish 0.5-meter Ebert spectrometer, which will resolve the S a doublet, has been used. For the case represented by Equation 24 the limit of detection could be greatly improved only by a n increase in path length (6) or a n increase in the stability of the source. However, from Equation 21, it may be seen that as the stability of the source is improved, other terms might become significant. The term ( ~ I , s / 1 " 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 I , the minimum detectable concentration may be calculated from Equation 24. The values of the parameters given in Table I were found as follows. The
Table I. Representative Results of limits of Detection for No o n d Cd in Two Flames"
Flame temp., "IC. ____ 2700 285W ~
Atomic line ?;a 5890 A . d S, (atoms/
3 2 12
CC.)
C, (p.p.m.)e Cd 2288 A4./ S, (atoms/
1 s x 0 034
cc.) C, (p.p.m.)c a
x
=
x
10-2, A j
=
1010 1 6 1 4
1010
x
10'0
9 4 x 105 0 017
1.
Stoichiometric H2/02 flame ( 2 3 ) with aqueous solution flow rate of about 2 cc./min., L = 0.5 cm. Stoichiometric C2H2/0, flame ( 2 3 ) with aqueous solution flow rate of about 2 cc./min., L = 1 cm. A V D = 4.1 X 1 0 9 set.-' at 2700O K., 4.2 X lo9 sec.-l at 2850" IC.; H ( 7 ' ) g go = 2, g. = 4; p = 0.97: A, = 5.89 x cm.; A i = 0.45 X 10* sec,.-l ( 4 ) ; 6 = 0.43. It is assurried that monochromator resolves 5890-A. line of S a doublet. e Experimental conditions assumed for converting from S, to C, values using Eq. 23 are as follows: F o r C P H 2 / O z flarne-n~ln~es= 1.2, Q = 132 cc./sec., e = 0.5, p = 0.04 for S a , and p = 1.00 for Cd. For H2/02f l a n i e - - n ~ / n=~ ~1.2, ~ Q = 125 cc./sec., e = 0.4, p = 0.10 for Na, and p = 1.00 for Cd. / A v o = 4.6 X 109 see.-' at 2700" K., 4.7 X 109 sec.-l a t 2850" K.; H(1') g gcJ = 1, gu = 3;
em.; A t 6 = 0.43.
=
p = 1; A, = 2.29 X 4.0 X 108 set.-' ( 4 ) ;
Doppler half width, Av,, can he obtained from the eupres4on given by Mitchell and Zemansky (141
7.15 X 1 0 - i ~ o
(25)
where R is the gas constant in ergs/ mole OK., c is the speed of light in cm./ second, T is the absolute temlierature, and M ais the atomic weight in atomic mass units. L-niversal con.5tants have been evaluated in the right hand ],art of Equation 25. Values of 6 in teriiis of the damping constant a = ( A u ~ bn) ,, d r 2 (where A u , is the Lorentz half width in second-') have tieen tabulated by foesner (15). The value of a for S a 5890 A . was obtained froin Hinnov and Kohn ( I O ) . Ailthough the value of a for Cd 2288 Ai.is not readily available, its value is estimated to lie about the same as that given for A\g 3281 X. by Hinnov and Kohn ( I O ) . Other listings of a values are to be found in tJnnies and Sugden (II)> Sobolev ( I Y ) , snd Mitchell and %emansky ( I d ) . .Ifuller discussion of the shalie of *i)ectral lines and its effect on atoniic abwri)tion is contained in .lj)l'endis 11. 'The ralculation of p is di,scii>sed in .li)l)entIis 111. VOL. 36, NO. IO, SEPTEMBER 1964
1949
Table II. Variation of
N,
with W for Several Monochromators and for Several Flame Types"
10-2 10-2 10-2 10-2 0.02 10-2 10-2 10-2 10-2 10-2 0.2 10-2 10-2 10-2 10-2 10-2 2.0 10-2 20 lo-: 10-2 10-2 1.1 x 10-2 10-2 200 101 . 2 x 10-2 10-2 4.5 x 10-2 1.1 x 2 8X a B = 1.3; M = 106; e , = 1.6 X coul: 1" = 8.9 X lo-' watt/cm.2 y = 5.3 x 104 amp/watt a t 400 rnp, 1.2 X 104 amp/watt at 600 mp; T , = 0 5 ; 4 = 0.005; s = RdW. 1 ern ; A = 25 cm 2 ; F = 50 cm.; x = * R d = 16 mp/cm. c Rd = 100 mp/cm. a t 400 mp. d Rd = 350 mp/cm. a t 600 mp. e I , = 1.3 X watt/cm.2 . ster. mp a t 400 mp (8). f I , = 4.0 X l o w 7watt/cm.2 . ster. mp a t 400 mp (8). watt/cm.2 ster. . mp a t 600 mp (8). I , = 1.0 X h I c = 6.8 X watt/cm.2 . ster. . mp a t 600 mp (8). (All values have been divided by the constant factor K 3 . )
The results in Table I are given in terms of atomic concentration and in terms of solution concentration (p.p.m.) for the experimental conditions listed a t the end of the table. These conditions are identical t o the ones used for similar calculations on the limits of detectability in atomic emission flame spectrometry (26). For the experimental conditions assumed, the limit of detectability for h'a was found to be about 0.5 p.p.m. for the CZHZ/OZand 0.4 p.p.m. for the Hz/02flame. These values are for flames of considerably shorter path length than normally used-Le., L = 0.5 cm. for Hz/Oz and 1.0 cm. for CzH2/Oz. Of course, if the same sample introduction efficiencies and flame temperatures can be maintained as the solution flow rate is increased to maintain the same effective atomic concentration, then there should be a corresponding decrease in the limit of detection in terms of solution concentration but not in terms of atomic concentration as the path length of the flame gases is increased. Fuwa and Vallee (6), however, obtained great sensitivities not so much by increasing the path length as by increasing the residence time of the atoms in the path of the radiation. The values for Cd are again, just as in the atomic emission paper ( M ) , much too low for a 0.5-cm. Hz/O2or a 1.0-cm. CzHz/O2flame. This seems to provide additional proof t h a t the choice of p (the fraction to account for incomplete compound dissociation and atomic losses due to ionization) is in appreciable error. Just as in the previous paper (26), if 6 is chosen to be 0.05, values quite close to those listed by Allan ( 2 ) and Gatehouse and Willis ( 7 ) are obtained. This again indicates that Cd undergoes fairly extensive compound formation in the two flames. This also indicates the futility of trying to apply the theory presented here to the calculation of limiting detectable solution
1950
ANALYTICAL CHEMISTRY
10-2 10 -2 10 - 2 10-2 ster.; H =
concentrations until more accurate data for the parameters become available. I n Table I1 the shape of the Y, us. slit width, W , curve is illustrated for several sets of conditions. Because the shape of the N , us. TI.' curve is independent of the spectral line, it is not necessary to evaluate all factors in Equation 21. The terms outside the square root have been combined in a constant term K., and Equation 21 has been rewritten in the form
xzAf
+ (Fr.Rd/Io)zWzAf]l'z(26)
where s has been replaced by RdTV from Equation 16. I n Equation 26, it is assumed that the monochromator isolates a single sharp line. In these calculations typical values of B, X , and y are taken for a 1P28 photomultiplier, and I" is estimated to be approximately the value given by Crosswhite (5) for a typical line of a n iron H C D T . The values of x and 4 have been estimated as shown in the table, and I , has been taken from the work by Gilbert ( 8 ) . For the chosen conditions, T V would have to be unreasonably small before the first term in Equation 26 would become significant. For this reason, Y,,, shows no increase as the slit width is decreased below the calculated optimum value (IV, = 0.19 cm.). However, slit widths below 0.19 em. will result in a smaller photodetector signal and correspondingly greater chance for electronic noise interference. A S the slit width is increased beyond the optimum value, N , increases very slowly for the conditions chosen, due to the increase in the last term in Equation 26, and very wide slits could be used before the increase would become significant. I n 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. I n such instances interference filters can be used with excellent results. If a single sharp line is isolated by the monochromator, then the only real difference between the Jarrell-Ash and the Beckman DU as atomic absorption monochromators is 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 a t 600 mp than a t 400 mp by about 6-fold for a n oxyacetylene flame, but the reciprocal linear dispersion of the DL is about 3.5 times higher a t 600 mp than a t 400 mp. The result is a slower increase in N, a t 600 mp than a t 400 mp for the oxyacetylene 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 Y, with It' a t 600 mp than a t 400 mp. The effect on .V, of varying other factors than those discussed above can be determined by examination of Equation 21. I n Appendices 11, I11 and IV some of the details of the preceding derivations and assumptions are discussed to a greater extent. APPENDIX
A
I.
DEFINITION A N D UNITS OF SYMBOLS
effective aperture of the monochromator, cm.2 = 1 - 11/10 = absorbance = transition probability, second-1 = damping constant = factor characteristic of photodetector surface = partition function = solution concentration, moles/ liter = limiting detectable solution concentration, nioles/liter (p.p.m.) = speed of light, 3 X 1010 cm./ second = angular dispersion, radians/mp = excitation energy of state u,e.v. = base of natural logarithms = electronic charge, 1.6 X coulomb = focal length of the collimator, cm. = frequency interval over which the amplifier-readout system responds, second-1 = gain per stage of the photodetector = statistical weight of state i = slit height, em. = Planck's constant, 6.6 X erg-second = transmitted intensity, watts/cm.Z ster. =
signal due to it,amperes source intensity, watts/cm.2 ster. = signal due to .I0, amperes = r.m.s. fluctuation in Io, watts/ cni.2 ster. second-1'2 = noise signal due to fluctuations Io, amperes = intensity of the flame continuum, watts/cni.2 ieter. mM = signal due to IC,amperes = r.m.s. fluctuation in I,, watts/ cm.2 ster. m p = noise signal due to fluctuation in I,, amperes = dark current (of t,he photodetector, amperes = intensity of thermal emission of the line of interest, watts/cm.2 ster. = signal due to I,, amperes = r.m.s. fluctuation in I,, watts/ cm.2 ster. serond-lJ2 = noise signal due t o fluctuation in I., amperes = intensity of fluorescent emission of the lines of interest, watts/ cm.2 ster. = signal due to I f , amperes = noise signal due to the photodetector, amperes = signal due to scattered incident radiation, amperes = noise signal due t o fluctuation in scattered incident radiation, aniperes = total noise signal, amperes = constant which. includes all terms outside the square root in Equation 21, a.toms/cc. = Boltzmann constant, 1.03 X 10-19 cc.-mm. of Hg/"K. in Equation 36 = atomic absorption coefficient at the line center, ern.-' = atomic absorption coefficient at the line center for a pure Doppler broitdened line, cm.-l = atomic absorption coefficient at the line center a t the minimum detectable concentration, cm. -l = at,omic absorption coefficient at v other than yo, cm.-l = path length through flame, em. = amplification factor of photodetector = atomic weight, atomic mass units = effective molecular weight of foreign species, atomic mass units = total atoniic concentration of species of interest, atoms/cc. = ground state concentration of species of interest, atoms/cc. = concentration of foreign species, particleslcc. = total niininium ,detectable atomic concentration, atoms/cc. = ground state minimuni detectable concentration, atoms/cc. = moles present a t temperature T nZgg and temperature 298" K., respectively = pressure of species of interest, mni. = Pressure of foreign species, = flow rate of unburned gases, cc./ second = gas constant, 8.3 X 107ergs/mole "K. =
Rd
=
s sC sm
T TI
reciprocal linear dispersion, mp/ cm. = spectral slit width, mp = spectral slit width at infinitely narrow mechanical slits, mp = spectral slit width as determined by the mechanical slit width, =
the pressure shift due to collisional effects is small) the resulting atomic absorption coefficient as a function of frequency can be expressed by
mp
= =
absolute temperature, "K. transmission factor
where
W = slit width, cm. Wo = optimum slit width, cm. 2A Y =-dK2 AVO
ZR
=
ZL
=
CY
=
p
=
ri y
= =
A
=
6
=
6~
=
c q(
=
X Xo
=
=
Y
= =
yo
=
AVD = AYE =
AVL = AUN =
number of resonance broadening collisions/seconds .atom number of Imentz broadening collisions/seconds 'atom ratio of emission line width to absorption line width fraction to account for incomplete compound formation and atomic losses due to ionization abundance of isotope i detector sensitivity factor, amperes/wat t a variable distance from the point v - y o , seconds-' factor t o account for line broadening other than Doppler broadening taken a t v = yo factor to account for line broadening other than lhppler broadening a t v other than yo efficiency of atomization ratio of intensity of hyperfine component i to the sun1 of the intensities of all other hyperfine components due to nuclear spin wavelength, cm. wavelength a t line center, cm. frequency, second-' frequency at absorption line peak, second-' Doppler half width of a line, second-' resonance half width of a line, second -1 Lorentz half width of a line, second -1 Natural half width of a line,
The parameters k , and k,* are, respectively, the atomic absorption coefficients a t frequencies Y and vOI where Y,, is the peak of the absorption line. The terms A Y N , A V L , AVO, and A Y E are, respectively, the half widths in second-' of the absorption line due to natural, Lorentz (due to collisions of absorbing species with foreign atoms, molecules, or particles), Doppler, and resonance (due t o collisions of absorbing species with other similar species) broadening. The term Y - y o represents the frequency distance from the center of the line and A represents a varia,ble distance from the point Y - v,. The term a is called the daniping ronstant and indicates the relative contribution to the broadening of a spectral line by the damping factors (natural, Lorentz, and resonance broadening). The term on the right of Equation 27 is the correction factor 6 ( o r 6") discussed in the body of the paper (Equation 20). The term k" (or k,") used in the body of the paper is k , at v = y o , and 6 is ( a / ~ ) a2
A
= 3.14 = correction
x
factor to account for hyperfine structure of source and absorption line = cross section for resonance broadening, = cross section for Lorentz broadening, cm.2 = lifetime of an excited state, seconds = flow -~ rate of solution, cc./minute = A I O / I o , second1i2
wi
= rtTi
OR
UL
r
+
APPENDIX 11. SHAPE OF SPECTRAL LINES AND EFFECT OF SOURCE LINE WIDTH
When the broadening of a spectral line ( 1 4 ) is due to contributions from a Gaussian shape-;.e., the Doppler effectand from an independent shape-i.e., the daniping effect due to natural daniping and other damping effects-then (if
(0
-
k"
at
v =
yo.
Thus
Y)2
(31)
k,*8
=
where
Likewise 6, is the value of ( a / ~ )x a2
p
+
v
+ (.
-
different from
k,
a t some frequency Y)2
and =
k,*6,
(33)
or, from Equation 31
k,
=
kQ,/6
(34)
The integration in Equation 27 is taken over the entire absorption line. The integral cannot be simply expressed and so nlust be evaluated ~ ~ a of n sa suitable series or by numerim1 integration. Poesner ( 1 5 ) has summarized the methods used for evaluating the right side of liquation 27 and gives a complete txhle of values of k,/k,* (or 6") as a function of u and a values. I n Pocsner's t:ihles the ternis H ( a , kus), H ( a , 0 ) , and k cmrrespond, respectively, to 6,, 6 , and ( V - V o ) / a V n in the nornenc~iatureused in this paper. The values of 2' a n d a depend on the evaluation of A V D , A Y I , , A Y . ~ and . AYR. The espression for A Y D has been given VOL. 36, NO. 10, SEPTEMBER 1964
1951
Table 111.
Per Cent Error in N and Absorbance Values Due to Finite Width of Source Emission Line as Compared to Width of Absorption Line a = AVEjAVD
k" L 0 . 00
0.25 0.0 3.0 3. 0
0
0 . 23 0 . 50
0.0 0.0 0.0
4.0 4.5
0 0 0.0 0.0
0.50
0.0 10.0
10.5
14 15 16
0.75 0.0 20.7 21.2
1 0 0.0 30 31
2.0 0.0 57
42 47 49
72 76 78
APPENDIX Ill. EFFECTS OF HYPERFINE STRUCTURE ON MEASURED ABSORBANCE
The effect of hl-pertine structure on atomic ahorpticin has been considered by \.idale ( L O ) . Hyperfine structure is due t o the nuclear spin effec*tarid isotope shift. \\'hethey due to the occurrence of isotopes or t o nuclear spin, the intensities arid absorbances of the individual hypertine components al\r.ays o c i w in a certain ratio ( 1 4 ) . If the effect o f isotopes alone is considered, then the ratio of line intensities of the source and absorbances due to atomic vapor is dependent c)nly on the ratio of nbundances. I 1 ~ : I 2 0 : I 3 O :, , ,
in the body of the paper (Equation 2 5 ) . According to LIitc hell and Zeniansky ( 1 4 ) , the Iment7 half nidth, A V I , is given by AVL =
z,
K =
(2 7 r ) U L 2 S , [2*RT(1 Alfa 1 Alf,)]"* (35)
+
where Z L is the nurnber of broadening collisions per second per absorbing atom, u(, is the ef'fective cross section for Lorentz broadening, in c m . 2 1 S, is the number of foreign gas particales/cc., and .Ill is the eft'ec>tive molecular weight of the foreign gas molecules. Assuming that the species in a flame are essentidly an ideal gas-i.e., PI = pressure of foreign particles = k.V,T, where k is the Boltzmann constant,-then .I-jis given by N j
Pj,'kT
=
=
9.74 X 10"Pj,'T
(36)
if the value of k is substit,uted and if P j is in nini. pressure. Substituting .Vj into Equation 35 and evaluating all constants give AVL. = 1.38 X
+
.'TI ( i l ; x a
1 0 2 3 ~ ~ 2 ~ , [ ( 1
l , ~ J I j ) ] l(37) ~* Hinnov and Kohn ( I O ) have evaluated a number of U L anti a values for several spectral lines of a number of atoms. Resonance broadening is due t o collisions between atoms of the same kind, and can be treated in the same manner as the 1,orentz tiroatiening.-i.e., replacing ZI, hy zR1 A U L by A V R j U L 2 by U R ' , P j by P o , 5 , by .Y, and .\I, hy .I/,. If this is done, the resonance half width is given by ( f i ) . AYR =
(2 'T)uRzs [2aRT(1'-Ifa 1 'A1fm)]'12 (38)
ZR
'7 =
+
probability in second-' of the spontaneous emission process. In the above discussion and equations it was assumed that the spectral lines were (iaussian in shripe. However, in many esperinients, spertral lines will show evidence of being :isyninietrical and of being shift,ed toward t,he red region. Therefore, the value of the measured atomic absorption c,oeffirient will be sonieKhat less than the value at Y = y o , the absorption line peak. In inost c':ises the deviation i n the atomic. absorption cfoefficient at v = y o f r o m the rnlculated value if shift arid usyniriietry were negligible is srnaller than the errors in the evaluation of the failtors in the expression for k" (LO). Also the broadening of a spectrnl line by the hyperfine structure results in an even sniialler error. Thus the fartor of shift and :tsynirnetry can usually he neglected. LIitc>hell and Zernansky ( 1 4 ) discuss the evnluation of the pressure shift of spectral lines of Hg and S a . For :t variety of foreign gases the pressure shift was seldom greater than 20"; of the 1,orentz half width for both S a and Hg. For all oalcnlittions performed in this paper the effects of pressure shift and asymmetry have been assumed negligible. The effect of the finite witlt,h of the hollow cathode line has also been neglected, arid it is interesting to investigate the niagnitude of the error that this might introduce. Table I11 shows the per (lent error in .Y and d o values as a function of the produc>t k"/, and the ratiolvE/AvD, the emission line width to the absorption line width. This ratio hns been designated a . The data under a equaling 0.25 and 0.75 were taken from 1-itlale ( 2 0 ) . The rest of the table was calculated from data given i i i .Ippendix I V of 1Zitc.hell and Zeniansky ( 1 4 ) on values of A, [where :la = 1 - ( f t / I o ) ] as a function of 01 and k " L . The per cent error in .V was taken as
and so
T h e natintl broadening half width is given f 14 t)y
Aw,- =
>(27rT)
=
.1, 2a
(40)
where T is the lifetime of the excited state in seimoiids. and .1 is the transition
1952
ANALYTICAL CHEMISTRY
Table I11 shows that the error in S increases rapidly as a increases, hut, with the precautions usually taken in atomic :rbsorpticin flame spertrometry, 01 should not exceed O.sj. For S a 5800 A . at 27700" I "The Spect,rum of Iron I, Johns Hopkins Spectroscopic Rept. S o . 13, .4ugust 1958. (6) Fuwa, K., Tallee, B. L., Asar,. CHEM.35, 942 (1963). ( 7 ) Gatehouse, R. M., Willis, J. B., Spectrochim. Acta 17, 710 (1961). ( 8 ) ,Gilbert, P. T., Jr., Symposium 011 Spectroscopy, Am. SOC. Testing Materials, Spec. Tech. Puhl. 269, 1960. (9) Goldman, S., "Frequency Analysis, &lodulation and Noise," McGraw-Hill, Yew York, 1948. 110) Hinnov. E.. Kohn. H.. J . Ont. SOC. Am. 47, 156 (1957) (11j James, C . G Sugden, T. 31 , .Vatwe 171, 428 (1953). (12) Jones, IT. G., Walsh, A., Spectrochzm. .Icta 16, 249 (1960). 113) Kusch. P.. Taub. H.. Phus Rev. ' 75, 1477 (1949). (14) hIitche11, A. C. G., Zeniansky, M.W., "Resonance Radiation and Excited Atoms,' University Press, Cambridge, 1961. (15) Poesner, 11. IT., Australian J . Phys. 17, 184 (1959). (16) Pringsheim, P., "Fluorescence and Phosphorescence,'! Interscience, S e w York, 1949. ( 1 7 ) , Rodda, S., "Photoelectric Mult,ipliers," Macdonald, London, 1953. (18) Sagalyn, P. L., Phys. Rev. 94, 835 119543. (19j Sobolev, S . M., Spectrochim. Acta 11, 310 (1957). (20) \.idale, G. L., U. S. Dept. Comm., Office Tech. Serv., P . B. Rept. 148, 206, 1960. 121'1 White. H. E.. Eliason. .4.Y .,. Phvs. " Rez,. 44, i s 3 (1939j. (22) Winefordner, J. D., Mansfield, C. T., 1-ickers, T. J., ilxal,. CIIEM.35, 1607 f 1963). ( 2 3 ) Ibzh., p. 1610. (24) Wnefordner, J. I)., Staab, R . A , , Ibzd., p. 165. ( 2 5 j Rinefordner, J. D., Vickers, T. J., Ibzd.. D . 161 11964). (26) Idid., p. 1939. ,
,
I
j - _ . -
RECEIVEDfor review January 31, 1964. Accepted June 12, 1964. Financial support from the Sational Science Foundation (SSF-G 19754) is gratefully acknowledged. Work was taken in part from a thesis by T. J . Vickers, submitted in partial fulfillment of the requirements for the Ph.1). degree a t the University Florida. cjf
