(5) Miller, H. W., Ibid., Rept. HW-22267, Oct. 1,5, 1851. (6) Pietri, C. E., Wenzel, A. W., AKAL. CHEM.35, 209 (1963). ( 7 ) Ryan, J. L., J. Phys. Chem. 64, 1375 (1960). (8) Wenzel, A. W., Pietri, C. E., U. S. Atomic Energy Comm. Rept. NBL-170,
(3) Ghiorso, A,, Jaffey, A. H., Robinson, H. P., Weissbourd, B. B., in “The Transuranium Elements,” G. T. Seaborg, J. J. Katz, W M. Manning, eds., NNES Div. IV, 14B (Pt. l ) , p. 1226, McGraw-Hill, New York, 1949. (4) KO, R . , U. S. Atomic Energy Comm. Rept. HW-57873, Oct. 20, 1958.
21 August 1961. (9r’Ibi$., R e p t . NBL-177, p. 35, March 1962. (10) Ibid., Rept. NBL-188, p. 51, December 1962.
RECEIVED for review March 25, 1964. Accepted June 8, 1964.
Calculation of the Limit of Detectability in Atomic Emission Flame Spectrometry J. D. WINEFORDNER and T. J. VlCKERSl Deparfment of Chemkfry, University o f Florida, Gainesville, Fla.
In this paper an expression relating the minimum detectable concentration in atoms per cc. of flame gases to the various experimental factors will b e derived for atomic emission flame spectrometry. Expressions will b e derived by which ihe optimum slit width for analysis may b e calculated, and the effect of slit width on the minimum detectable concentration will b e discussed. Also an equation for converting from atoms per cc. of flame gases to moleir per liter of solution introduced into the flame will b e derived, and samplte calculations will b e carried out to illustrate the use of the derived equations.
I
a large number of factors affect the results of
T IS WELL KNOWN that
atomic emission flame spectrometry. However, until the present time, no adequate treatment has been given describing the quantitative relationship of experimental factors to the sensitivity of analysis. The experimenter has been forced to rely on trial-and-error methods of determining optimum conditions which are time consuming and often misleading because of the large number of variables and their interdependence. I t may be that some of the contradictory observations that have been reported in the literature may be attributed to a lack of understanding of the effect of variations in experimental conditions. The present work is a n attempt to remedy this situation and provide information of aid in routine analysis as well as in theoretical considerations. The development of a limit of detection theory for atomic emission flame spectrometry was suggested by the work of Mandelstam and Nedler (17) who developed a theory for the 1 Present address, United States Army, Ft. McClellan. Ala.
limit of detection of atoms excited by a d.c. arc and by a hollow cathode discharge tube. THEORY ON THE LIMIT O F DETECTABILITY IN A T O M I C EMISSION FLAME SPECTROMETRY
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 in cm. are, respectively, w’ and H . The effective aperture of the monochromator is il, cm.*, and the focal length of the collimator lens or mirror is F , em. The term TIis the transmission factor of the spectrometric system (includes entrance optics as well as monochromator optics) as determined by absorption and reflection losses at all optical surfaces. The spectral slit width of the monochromator is denoted s and is given (3, 4, 16) approximately by 8 = , 8
9 a d 980
=
sm
$.
8c
(1)
where s, is the spectral slit width of the monochromator when using wide slits, sd is the spectral slit width of the monochromator when using infinitely narrow slits and is determined by the diffraction limit of the spectrometer assuming focusing aberrations are negligible, so 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 a t t h e exit slit and s, = sd sa. I n this paper the spectral slit width will always have units of mp. The term s, is mp is given (4, 16) by
+
,S
=
(l/DF)JY = 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 em.i.e., A. per mm. The term s d , in mp, is given (4, 16) by Sd =
( ~ / D ) ~ / c=u oFRdX/ao
(3)
where A is the wavelength setting of the monochromator in cm. and a0 is the width of the beam of radiation in cm. a t the dispersing element. The term s,, in mp units, is a constant characteristic of the monochromator. Equation 1 is not strictly correct because the above effects are not strictly additive ( 3 ) . 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 = *s, se2. I n this paper, the addition relationship for s, given in Equation 1, will be used. I n addition to the above requirements concerning the entrance optics and the monochromator, the following
+
VOL. 36, NO. 10, SEPTEMBER 1964
1939
requirements will be made for the system under consideration. I t 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 noire is a result of derector and flame noise. The above assumlitions and requiiwiients regarding the instrumental system to be used are generally valid for any good conimercial flame spectrometer; and, therefore, tlie following discussion should be directly applicable to many experimental systems. The mininiuni detectable concentration, S,,in atoms per cc. of flame gases will be defined as that concentration which produces a n average output current (anodic current of phototube) i,, in amperes, such that i, = 2 A i T (4 __
where AiT is the root-mean-square fluctuation in the background anodic current of the phototube. To simplify the use of the following equation, the units and definitions of all symbols are given in ;2ppendis I. The value of i, can be shown (see Appendix 11) to be given (IS) by
i,
=
tiyT,rmTVH((.4,:F*)n
1940
ANALYTICAL CHEMISTRY
where A is the wavelength setting of the monochromator and Xo is the wavelength of the line center. For nionochroinator wavelengths greater than A, s and less than ho - s, ti is zero. If the monochromator has sufficient resolution to isolate a single, sharp spectral line, then the value of K can be made equal to unity by adjusting the monochromator wavelength, A, to the peak wavelength of the spectral line,
+
AO.
The integrated intensity I , of a single, isolated spectral line in units of watts/cm.2 ster. [usually called steradiancy ( 2 0 ) ] is given ( 2 ) by the well known equation
(5)
where y is the photosensitivity factori e., the current in amperes produced a t the anode of the detector for each watt of radiant power incident upon the photocathode-and I,, is the total radiant intensity (integrated intensity) of the spectral line (in w a t t s ~ c n i .ster.) * produced by the minimum detectable concentration. The term A : F 2is the number of steradians viewed by the tem as long as the effective aperture is filled with radiation. This is true whether the source is viewed directly or a lens or niirror is used to focus a selected portion of the flanie on the slit. The parameter n is the number of solid angles of value .-t ‘ F * which are gathered into a single solid angle by means of a suitable arrangement of entrance optics. As has been pointed out by Gilbwt ( I O ) . it is possible by the use of a suitable system of mirrors to increabe the intensity incident upon the spectrometric system. Thus, if a niirror is suitably placed so as to focus a n additional image of the source on the slit, then the total incident intenqity is inR,,T,,, d i e r e R, creased by ti = 1 is the rrflectance of the mirror and T , is the transniittanc~eof the flame for the Iiarticular s l m t r a l line in concern. {-sing the siiectronietric Pyzteiii which is descritied at the beginning of this Pertion. a single viewing (72 = 1) of the flanie is aswinetl. Once a theory is worked out for a single viewing of the flame source, it i.s easy to evaluate n for any number of 1-ien-ings by proper consideration of the geometry of the
+
entrance optics. Gilbert (10) has considered several possible optical arrangements for increasing the light gathering poi\ er. The parameter K accounts for the position of the spectral line with respect to the slit function distribution curve. For equal entrance and elit qlits, the distribution curve is triangular, and the slit function parameter is given (4) by
I,
=
(10-’/4~)S,Lh~oAt [g,,lB( T )]e-Eu/kT
(7)
The above equation has been derived assuming that the flame is an optically thin medium, which is valid at the limit of detection (19), and assuming the flame source is in thermal and chemical equilibrium ( 1 8 ) ,which is approximately valid for the central region of the outer cone of most analytical flames. The term S, is the minimuin detectable number of atorm’cc. of flame gases, h is Planck’s constant in erg-seconds, the 10-7 is necessary to convert from ergslsecond to watts, uo is the frequency of maximum intensity of the spectral line in second-’, A , is the transition probability in second-’ for the transition from the upper state u to a lower state (usually the ground state of the atom, designated 0), E , is the energy of the upper state, k is the Boltzniann constant in units consistent with E,, T is the absolute temperature, and L is the average diameter or thickness in em. of tlie flanie region being focused on the monochromator entrance slit. I n Equation 7 it has been assumed that .Ymis given with good accuracy by SO,, the minimum detectable number of ground state atoms per cc. of flame gases. This assumption is valid for most atoms when using the flames normally ernl~loyedin analytical flame spectrometry. The st the upper state is gu, and the partition function over all states i q B ( T ) which is defined by
where the summation is carried out over all stat’es. However, only in cases in which t’here are excited states within 1.5 to 2 e.v. of the ground state-e.g., Cr at 3000’ K. in which B ( T ) is about 9yc greater than go-does B ( T ) differ significantly from go, and so in most cases the siniplification B ( T ) = go can be made with negligible error. -4s long as the spectral line is single, sharp, and isolated, the value of I , is given by Equation 7 . If radiation from more than one line of the same element is passed by the spectrometric system, then I , niust contain additional terms for each line similar to the right hand term of Equation 7 . In this case i,, will be given by an equation similar to Equation 5, namely,
i,
=
T,Tt”(S!F2)nCK,yjl,j
(9)
3
where t’he suniniation is over all spect,ral components passed by the exit slit. Because all components will not be passed centrally through the exit slit, ti1, t’he slit function for each spectral component passed by the exit slit, will be less than unity for each line. The phot’osensitirity factor, y i 2 for each spectral line passed by the exit slit will, however, be nearly the same as the photosensitivity factor for the wavelength a t which the monochromator is set as long as the change in y with wavelength is small over the spectral slit width of the instrunient. I n any event it would be a relatively simple matter to measure experimentally the value of y 1 for each spectral line passed by the exit slit or to determine y j approximately by means of the manufacturer’s spectral response curves for the photomultiplier tube being used. The evaluation of K ? for each spectral component can be performed by Equation 6. The greatest problem in evaluating the summation in Equat,ion 9 is in determination of the number of spectral components passed by the esit slit-Le., 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 xaveleng-ths near the extremes of the slit distribution. The intensity, I,,, can be evaluated if g., = I t , and E , for each spectral component are known. If the spectral line is so broad that the intensity read by the instrunient is not the total line intensity, then additional factors niust be included in Equation 9. This problem has been considered by Winefordner (22) and Brodersen ( 6 ) . However, most spectral
lines produced 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. I t is not possible to consider in a general way the case in which the sllectral line is not re:.olved froni lines of other elements because this would require a detailed knowledge of the matrix is to be performed. I n the following discussion: the spectral line will be considered single, sharp, and isolated, anti the monochromator wavelength a i l 1 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. I, d l be given by Equation i and i, will be given by i, = yTjI,TI*H.l P ( K = 1 and n = 1). As iioted in the above discussion. there are cases in which the assumption of a sin:;le: sharp: and irolated line is not valid. However, it should be pointed out that even if in a particular case a line does not meet these criteria, it is still possible to obtain exact rebults by substitution of the prol~er values into -the summation of Equation 9. More energetic sources such as arcs, sparks, etc., certainly do not meet the above criteria. The root-mean-square fluctuation current, A i T . is a result of the rootmean-square fluctuation current due to noise in the flame source, and the root-mean-square fluctuation current due to noise in the phototube, AiD. The flame noise at th,. limit of detection is 1)riniariIy due to fiuctuations in the intensity of the continuous background radiation, and the i h t o d e t e c t o r noise is due to the shot effect and thermal noise. The two noise signals add quadratically (12) so that is g i w n by
-zcl
iG
The value of
xpis given (21) by 4,EToAf 'Ro]"2
+
+
i,
=
yT/IClI-H(rl,'F2)s
(12)
where I , is the intensity per unit \\-avelength interval of the continuuni.e., I , has units of watts ' ~ i i i ster. .~ nip (called steradiancy per unit wivelength interval). If the spectral line teni. burh as 1Ig 2852 or Cd 3261, then the aliproljriate ex~iresbionfor the background intenhity, I,, must be used in Equation 12. Because background other than continuuin depends upon the flame coniiiouition and samlile tylle. this cannot be considered in general. The value of can also be shown to be given by (see .\lipendis 11)
-
yT,ZTI-H(;l 'F2)s(Lf)1 (13) where AI, 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 5,,one may write Ai,
=
The equation for 2, can be somewhat simplified because the thermal noise is generally (21) negligible when compared to the shot effect for most syqtems, and so Equation 11 becomes
--
Lip = [2e,BMLf(id
-
+ i,)]1'2. (15)
Substituting for Ai,,> i,: antl Ai, and collecting terms, the eslire.ssion for the minimum detectable concentration is obtained
I3y evaluating all factors in Equation 17 for the particular esperimental setup in concern: IfT0 can be found (use of a graphical method i h the simplest means), If the flame background is relatively low at the wavelength setting of the monochromator, then s,, i. much larger than s,, and s = s, = Rdli-. For this case lt', is given by a n expression considerably s:iml)ler than Equation 17, namely,
where Equation 18 ('an be found directly from Equation 17 by assuming s, is very small so that the secwnd and third terms berome negligible. I , and A I c increase [ A I c increase. ap~)roxiniately proportiona1l~-with I, (IO)],the last term become.. negligihlc when compared t o the other term.. Therefore, at , ol)timum slit large values of I C the width. will become quite small, and in fact it can hecome bignificantly smaller than the practical re.mlving power expressed in terms of slit width (13)-i.e.] smaller than the olitimum slit n-idth for maximum spectral resolution with little lo..< in intensity. Of course. for ver>- small slit widths, Equation 17 must be used. If t h e calculated slit nidth from Equation 17 or 18 is extremely >mall-i.e.. diffraction efferts due to the hlit are significant-then a gain in senbitivity re:ults if the exit slit width i. wider than the entrance slit nitlth to allow coml)lete collection of the diffraction ])attern hy the phototube. However. in this caye the equations ahovc must lie modified and the sliectral line must nece,arilybe single and isolated. Ikcause the ('ase of very imall dit width. i? rarely a practical cabe in analytical flame spectrometry and because conkideration of thi.: case would coiiiitleral)ly cwmplicate the theory. it wa? not done.
-
(11)
where e, is the electronic charge in coulombs, B is a constant approximately equal to 1 1 G 1 'GZ 1,/G3 . . ., where G is the gain per stage of the photomultiplier tube, .If is the total anil)lification factor of the phototube, and Lf is the frequency response bandwidth (12. 13) in second-' of the amlilifier-readout circuit. Both flame noise and phototube noise are assumed to be equally distributed over Lf-i.e., the noise is assumed to be white ( 1 2 ) . The dark current i n amliere> a t thc anode, ~ ~ r o d u c eby d thermionic emission, is id,and the signal in amlieres produced at the anode due tic) the incident intensity of continuous radiation from the flame i. i,. The teml)erature of the
+
load resistor in O K. is To and the load resistor of the detector circuit has a resktance of Ro ohms:. The value of i, in amperes (see pendis 11) can be shown to be given by Equation 12, namely,
+.
in units of atomslcc. of flame g a m . The constant 3.8 X 1 0 3 4 results when 8~ X 10:is divided by Planck'y constant. If in Equation 16 is differentiated (see dppendix 111) with respect to Ifand minimized. then the condition for is found optimum slit width in cm., llTo, to be satisfied when
e,RJI y T ,Hi.1
P) IcscTI-o2e,BJfid
=
0
(17)
Won-ever. some caution must be exercised when optimum .lit widths calculated froni 14:quatioii.
