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
CHROMATORDETECTOR 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 ROOTMEANSQUARE
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 SignaltoNoise 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 Signalto
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, electronvolts
ec electronic charge, 1.59 x 1019 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 1027 ergsec./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 rootmeansquare 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 ekoL)
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 (selfabsorption) 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.
rootmeansquare 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 rootmeansquare 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
anderror 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 signaltonoise 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 signaltonoise 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 atomizerburners, 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
amplifierreadout 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 rootmeansquare 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 (107/4[)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 107 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 + g2E2/kT.... eEi/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 rootmeansquare fluctuation current, AiT, is a result of the
rootmeansquare fluctuation current due to noise in the flame source,
Aic, and the rootmeansquare 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 amplifierreadout 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 rootmeansquare 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
atomizerburner 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 1022 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 airhydrogen and airacetylene flames.
The value of AIc is given by cI,, where y is the ratio of the root
meansquare 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
E4<
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
r4
CO
C4
cu
0
Le
4
I
0
0
0
a
4
co
4
\o
0
O
0
0
4
4
0
44
o
0
zo
*
.N
41
0
r4
41
O
0
:3
Cd
r44
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
aI C5
i0 01
0cd
> r~cc
40
a 0
14 44
n ) X
S 0
Q) 00
o in
H* ai
Ica4
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
s4
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
4i
0
0
a
i4
d
II
O
H1
II
II
~4
00
*
v40
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 rootmeansquare 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 peaktopeak 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 totalconsumption atomizerburner, 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 104 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
Pwas determined in the following way. The ionization constant for the
process Na  Na+ + e was calculated to be 1.01 x 107 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 109 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 airacetylene 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
atomizerburners 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
41 Q> (U) a :3
00 1 M 0 0o> C
a W up u o co u4
M < C ud l E4 c
4 0 O C00 00 P
0 n 0 0 000 0 *4
0 .a O 0 ) 4 40
H 0 oO 0 r 0 .
E4 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 444
 4 04 >iO 0 0 0
p3 [.0 U .* 0 0 4
HOM 0
S E 044 ( 00 4 Q 4 r 400
E 0 00 4 0 40 4 0
O4 , 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 4J4
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 Zi 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
.ED (U) 41 4 C0 e
H 0 C :.% C. l
O CIJr:. 0 O 3c 0 E
'0 *2 o > *6 o0
E4 W c4 L4 O 4J .4 *i0 u co
p S a CO 44 U )
(. 4 ',4 *O 0
SE 0 ( 0 C
)E4 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 * CEI ) C u
/3 U 4 a e bO 4 3 0b14
00 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 () r4 ( 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 fluorinehydrogen,
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 1019M[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 108K
1.5 x 108Kc
1.0 x 108Kc
0.90 x 108Kc
0.82 x 108Kc
0.78 x 108Kc
0.79 x 108Kc
0.91 x 108K
1.3 x 108Kc
2.6 x 108K
*The above calculations were performed using equation [22] and the
following data: id = 108 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 1011 watts/cm.2
ster. mp sec.
Using the above data and equation [18], Wo is 0.015 cm.
conditions: id = 108 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 109 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 108 (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 1012 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 amplificationreadout 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 rootmeansquare 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 rootmeansquare 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 YTfWH(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., epkOmL
1 pkmL, one may write
TfWH(A/F2)IokomL = 2ZiT [30]
The rootmeansquare 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
ip2 = 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 rootmeansquare 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
iT = 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 overall 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 JarrellAsh 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 = 102, 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 105
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 105
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 backgroundincreases, 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 107 ~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
JarrellAshb DUC Dud
H2/02e
102
102
102
102
102
C2H2/02f
102
102
102
102
1.2 x 102
H2/02
102
102
102
102
102
C2H2/02
102
102
102
1.1 x 102
4.5 x 102
H2/02g
102
102
102
102
1.1 x 102
aB = 1.5; M = 106; e 1.6 x 1019 could ; 1 = 8.9 x 104 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= 102; =
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 108 watts/cm.2 ster. mp at 400 mp.
flc = 4.0 x 107 watts/cm.2 ster. mp at 400 mp.
gIc = 1.0 x 108 watts/cm.2 ster. mp at 600 mp.
hic = 6.8 x 108 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
102
102
102
102
2.8 x 102
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 JarrellAsh 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 6fold 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 monochromatordetectorreadout
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 signalto
noise ratios, i.e., differentiating the signaltonoise ratio with
respect to W, equating to zero, and solving for the optimum slit width,
Wo. The signaltonoise 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 noisetosignal 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 signaltonoise ratio for atomic absorption flame spectrometry
is obtained by dividing equation [49) by equation [50].
(io it)/AIT [54]
YTfWH(A/F2)IO(1 ePkoL)
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 noisetosignal
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 ekOL)
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 fueltooxygen 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 signaltonoise ratio over the temperature range which that flame
can exhibit. The calculation of N and the signaltonoise ratio then
will be repeated for other flames. For both atomic emission and atomic
absorption flame spectrometry, plots of signaltonoise 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.3atm./OK. (1.38 x 1022), 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]
FKK3K2 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.3atm./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 signaltonoise 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
E4
.4
,4
.4
*I
Hr
^0
I
I
x
I
.
\
0
i4
I
x
O
I *
1 <
01
/
/
x
.4
01
0
i4
x
.4
*4
I
/
/ x
/ ^
4J
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
S4 <0 I
i 
I
o
41
0
E4
sI
sI
el
0
(a
iCl
J
cr
m
U)
U)
H
4
0
a o
En
0
r.
.re
t0
,
0U
U
ri
oc'
CO
'4
0
r4
,4 U
4U2
C 04
CO
4J 51
flame conditions the instrumental parameters, except W, will remain
constant because the object sought is the flame type which will give
the largest signaltonoise 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 NAtLeEu/kT [77]
1074B (T)
where all the terms have been previously defined. In the high con
centration (or selfabsorption) 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 IIn 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)1/2L1/2N1/2eEukT [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 NLeEu/kT( g + 2g2) [90]
i= 80,1c) 1074cB (T) eigl1+1 K292) 90]
Collecting constant terms gives
i = kgkll( c)1/2NLeEu/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/2N1/2LeEukT [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 V12 (~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 rootmeansquare 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 atomizerburner.
Because AiT is approximately the same for all three flame types,
the flame type giving the maximum signal will also give the maximum
signaltonoise ratio. For low concentrations where the signal, i, is
given by equation [80], the signaltonoise ratio is given by
i/Ai = kclIc)/2LNeEuT [97]
TABLE 6
OPTIMUM SLIT WIDTH AND TOTAL ROOTMEANSQUARE NOISE
FOR THREE FLAMES
Flame Wo (cm.) AiT (amps.)
H2/air 6.1 x 102 3.2 x 1010
H2/02 1.7 x 102 3.1 x 1010
C2H2/02 6.5 x 103 3.2 x 1010
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 1019 coulombs,
i = 107 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 1010
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 108 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 signaltonoise ratio is given by
i/iT = kc c)1/2LNil/2Nl/2eEu/kT 9[8]
For the case of the unresolved doublet spectral line (equations [91]
and [95]), at low concentrations the signaltonoise ratio is given by
i/AiT = kd(c)1/2NLeEu/k [99
where kd = k8kll/iT, and at high concentrations the signaltonoise
ratio is given by
i/ai = kd ) 1/2i/2N1/2LeEukT [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/iTkc is calculated from
equation [98. At low concentrations i/AiTkc = i/iTkd
(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 signaltonoise
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 signaltonoise 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
14
4
0
4
xl
CO
'I
C
SI I I r
0
0
c1
u
0 <'
4l
0
J0
I4
o
*4
3
00
1I
oo
1
o0
Zre
0
L
U0
inm
t
0
" EW
,4 1
o .U
40
4 C/
0
C'I 41
cfE
At high concentrations the plotted signaltonoise 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, JarrellAsh 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 signaltonoise ratios i/AiTkc or i/AiTkd depend only on the one
half power of Ni.
Neglecting the small instrumental dependence of the signalto
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 selfabsorption region. Above this line the signal, and hence the
signaltonoise ratio, is proportional to the squareroot 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/iTkc 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 monochromatordetector combination, and
therefore this line must be calculated for specific cases.
In the region between the selfabsorption line (S.A.) and the
limit of detectability line (L. of D.), the spectral line is detectable
and selfabsorption of radiation is negligible. In the H2/02 flame at
25000K. the point at which selfabsorption becomes important corresponds
to a PT value of approximately 4.4 x 106 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 103 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 1011 to 106 atmospheres (or solution concentrations of
approximately 5.2 x 109 to 5.2 x 104 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 selfabsorption
for PT values above approximately 4 x 106 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 signaltonoise 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 epkOL
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],
(Ics/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 chambertype
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 ePkoL) [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 halfwidth, 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 ek4NL/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 3T8 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 ekl5NL/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 signalto
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
El
M l1 l
O
0
C)
0
en
o
o
o
co
N
O
0
O
4W
C)
Sli
o o
0 0
co
N
00
Cd
O 0
r4
o
0
0
N
0
0
0
0
0
o
4d
0
0
4
O^
0
.4
C)
co
c4
S1
0
EH
0
r4
0
Cd
0
0
4d
4
)
E4
z
0
0)
0
ca
a)
Cd
i
ed
O
0
I
analysis. The absolute value of the signaltonoise 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 100625e /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 109 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 1031c, 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 107 2.5 x 1015 1.3 x 107
2000 9.1 x 105 6.4 x 1012 4.1 x 105
2400 3.2 x 103 1.5 x 109 2.0 x 103
2800 4.2 x 102 7.4 x 108 3.1 x 102
3200 1.4 x 101 1.6 x 106 2.5 x 101
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 LandoltBornstein 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 signalto
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"!
*oI o *o ,,
om I II
1 0 C) o
c 00 0
I 6 0 0
f O0 U r0' 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 atomizerburner, 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 101
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 102, 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 selfabsorption 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 atomizerburners. 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 "fuelrich" 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 signaltonoise 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
signaltonoise ratios of the various flames must be related to each
other in th same way as the theoretical signaltonoise 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 "fuelrich" flame. The "fuelrich"
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 peaktopeak
noise was measured at several slit widths above and below the calcu
lated optimum slit width. The peaktopeak noise was multiplied by
1/2 42 to obtain rootmeansquare noise, and the ratio of W to root
meansquare 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
signaltonoise 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 signaltonoise ratio versus
W plots similar to the H2/02 flame plot. A Moseley xy 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 SignaltoNoise 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 thework 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 selfabsorption 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 signaltonoise 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
trialanderror 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
nonthermal, 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 CurrentIntensity 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].
