Group Title: calculation of limits of detectability and optimum conditions for atomic emission and absorption flame spectrometry
Title: The calculation of limits of detectability and optimum conditions for atomic emission and absorption flame spectrometry
CITATION PDF VIEWER THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00097950/00001
 Material Information
Title: The calculation of limits of detectability and optimum conditions for atomic emission and absorption flame spectrometry
Physical Description: xii, 124 l. : illus. ; 28 cm.
Language: English
Creator: Vickers, Thomas J., 1939-
Publisher: s.n.
Place of Publication: Gainesville
Publication Date: 1964
Copyright Date: 1964
 Subjects
Subject: Absorption spectra   ( lcsh )
Radiation   ( lcsh )
Flame photometry   ( lcsh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis - University of Florida.
Bibliography: Bibliography: l. 121-123.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00097950
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000423912
oclc - 11026624
notis - ACH2317

Downloads

This item has the following downloads:

PDF ( 4 MBs ) ( PDF )


Full Text











THE CALCULATION OF LIMITS OF
DETECTABILITY AND OPTIMUM CONDITIONS
FOR ATOMIC EMISSION AND ABSORPTION
FLAME SPECTROMETRY








By
THOMAS JOSEPH VICKERS


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
April, 1964















ACKNOWLEDGMENTS


It is with sincere pleasure that the author takes this opportunity

to acknowledge his debt of gratitude to his research director, Dr. J. D.

Winefordner. The author is indebted to Dr. Winefordner for his instruc-

tion, encouragement, and advice, and many hours of his time spent in

fruitful discussion of the material presented in this dissertation.

The author also wishes to express his gratitude to Rev. William

J. Rimes, S. J., of Spring Hill College. It was his assistance and

dedicated teaching which encouraged the author to pursue higher studies

in chemistry.














TABLE OF CONTENTS


Page


ACKNOWLEDGMENTS

LIST OF TABLES

LIST OF FIGURES

KEY TO SYMBOLS

Section

I. INTRODUCTION

II. CALCULATION OF THE LIMIT OF DETECTABILITY FOR
ATOMIC EMISSION FLAME SPECTROMETRY

Introduction
Derivation of Equations
Discussion and calculations

III. CALCULATION OF THE LIMIT bF DETECTABILITY FOR
ATOMIC ABSORPTION FLAME SPECTROMETRY

Introduction
Derivation of Equations
Discussion and Calculations

IV. CALCULATION OF OPTIMUM CONDITIONS

Introduction
Selection of Optimum Slit Width
Effect of Flame Conditions on Atomic
Concentration
Optimum Flame Conditions for Atomic Emission
Flame Spectrometry
Optimum Flame Conditions for Atomic Absorption
Flame Spectrometry
Calculations
Experimental Verification of Theory


V. CONCLUSIONS

APPENDICES

LITERATURE CITED

BIOGRAPHICAL SKETCH


124














LIST OF TABLES


Table Page

1 REPRESENTATIVE RESULTS IN ATOMIC EMISSION FLAME
SPECTROMETRY FOR LIMITS OF DETECTABILITY IN
NUMBER OF ATOMS PER CM.3 OF FLAME GASES FOR
TWO ELEMENTS (Na and Cd) IN TWO FLAMES AND FOR
TWO MONOCHROMATORS WITH OPTIMUM SLIT WIDTHS 17

2 CALCULATED VALUES OF OPTIMUM SLIT WIDTHS IN CM.
AND LIMIT OF DETECTABILITY IN ATOMS PER CM.3
FOR SEVERAL ELEMENTS WITH RESONANCE LINES IN
WIDELY DIFFERENT SPECTRAL REGIONS AS A FUNCTION
OF FLAME TEMPERATURE FOR A REPRESENTATIVE MONO-
CHROMATOR-DETECTOR SETUP 21

3 VARIATION OF Nm WITH W FOR A PRISM MONOCHROMATOR
AT 420 mp WHEN USING AN OXYHYDROGEN FLAME 24

4 REPRESENTATIVE RESULTS IN ATOMIC ABSORPTION FLAME
SPECTROMETRY FOR LIMITS OF DETECTABILITY IN
ATOMS PER CM.3 OF FLAME GASES FOR Na AND Cd
IN TWO FLAMES 36

5 VARIATION OF Nm WITH W FOR SEVERAL MONOCHROMATORS
AND FOR SEVERAL FLAME TYPES 39

6 OPTIMUM SLIT WIDTH AND TOTAL ROOT-MEAN-SQUARE
NOISE FOR THREE FLAMES 62

7 VALUES OF Kl, K2, AND K3 AS A FUNCTION OF T 74

8 PER CENT ERROR IN N AND ABSORBANCE VALUES DUE TO
FINITE WIDTH OF SOURCE EMISSION LINE AS COMPARED
TO WIDTH OF ABSORPTION LINE 96

9 HYPERFINE STRUCTURE COMPONENTS OF THE Na 5890 A.
LINE 101













LIST OF FIGURES


Figure Page

1 Calculated Plots of Atomic Concentration of Sodium
Versus Flame Temperature for Several Total
Pressures of Sodium and for Several Flame Types. 54

2 Calculated Plots of Signal-to-Noise Ratios for the
Na 5890, 5896 A. Doublet Versus Flame Temperature
for Several Total Pressures of Sodium and for
Several Flame Types. 64

3 Calculated Plots of NL/T1/2 Versus Flame Temperature
for Several Total Pressures of Sodium and for
Several Flame Types. 70

4 Calculated and Experimental Analytical Curves for
the Na 5890, 5896 A. Doublet for Several Flame
Types. 76

5 Calculated and Experimental Plots of Signal-to-
Noise Ratio Versus Slit Width for Sodium in a
Stoichiometric H2/02 Flame. 82














KEY TO SYMBOLS


A effective aperture of the monochromator, cm.2

a damping constant, no units

A absorbance, no units

AT total absorption of spectral line, no units

At transition probability, sec.-1

Aa 1 I/I1, no units

A9 atomic absorptivity at frequency V, no units

B factor characteristic of photodetector surface, no units

Be rotational constant of molecule cm.-1

B(T) partition function, no units

C concentration, moles/liter

c speed of light, 3 x 1010 cm./sec.

Ci constant defined in text

D angular dispersion, radians/mp

DNaCl dissociation energy of NaC1, electron volts

e base of natural logarithms, no units

Ei energy of state i, electron-volts

ec electronic charge, 1.59 x 10-19 coulombs

ef expansion factor, no units

F focal length of collimator, cm.

Af frequency response band width, sec.-

G gain per stage of photomultiplier, no units








gi statistical weight of state i, no units

gol statistical weight of ground state of ion, no units

go* statistical weight of ground state of molecule, no units

H slit height, cm.

h Planck's constant, 6.62 x 10-27 erg-sec./atom

I integrated intensity, watts/cm.2 ster.

i output signal, amperes

Io transmitted intensity with no sample in flame, watts/cm.2 ster.

iO signal due to IO, amperes

AIo r.m.s. fluctuation in Io, watts/cm.2 ster. sec.-1/2

iTo noise signal due to fluctuations in I1, amperes
2
Ic intensity of the flame continuum, watts/cm.2 ster. mp

ic signal due to Ic, amperes

ZIc root-mean-square fluctuation of intensity of flame continuum,
watts/cm.2 ster. mp sec.-1/2

Zic noise signal due to fluctuation of flame continuum, amps.

id dark current, amps.

Ie intensity of thermal emission of the line of interest,
watts/cm.2 ster.

ie signal due to le, amperes
2 -1/2
AIe r.m.s. fluctuation in Ie, watts/cm.2 ster. sec.-

Aie noise signal due to fluctuation in Ie, amperes

If intensity of fluorescent emission of the lines of interest,
watts/cm.2 ster.

if signal due to If, amperes

ii chemical constant of species i, no units

Im intensity due to N watts/cm.2 ster.

im output signal due to Nm, amps.

Zip phototube noise signal, amps.


vii








is signal due to scattered incident radiation, amperes

AIS r.m.s. fluctuation in intensity of scattered radiation,
watts/cm.2 ster. sec.-1/2

Lis noise signal due to fluctuation in scattered incident
radiation, amperes

It transmitted intensity with sample in flame, watts/cm.2 ster.

it signal due to It, amps.

AiT total noise signal, amps.

I(B intensity of a black body radiator at frequency V watts/cm.2
ster.
B
I o intensity of a black body radiator at frequency Vo,
watts/cm, ster.

k Boltzmann constant, see units in text

kO atomic absorption coefficient at the line center, cm.-1

k0m atomic absorption coefficient at the line center at the
minimum detectable concentration, cm.-1

ko* atomic absorption coefficient at the line center for a pure
Doppler broadened line, cm.-1

Kc constant defined in text

KH dissociation constant for HC1, atm.

Ks constant defined in text

kc k8 kg/AiT

kd k8 kll/AiT

K1 dissociation constant for NaC1, atm.

K2 ionization constant for Na, atm.

K3 dissociation constant for NaOH, atm.

kI CTfH(A/F2) nI

k2 2BMecAf

k3 YTfH(A/F2) Ic

k4 YTfH(A/F2)ic

k5 TfH(A/F2) Io (1 e-koL)
viii








kg Tf H(A/F2)

k7 (2BMecI)/( T HA/F2)
7 2B^cid 1
k8 ITfH(A/F n (TH(A/F2) (Af)l/2 R)2
h Vogu
k9 47 x 107B(T) At
hV 2 VD gu 1/2
( At a)
10 107 c g rnn2 B(T)

h A o At
11 lo 0 4B(T) (11 22)

h 2 2 AVD a At 1/2 /2]
k12 107 c ,-ln2 B(T) L 1 (gl) 2(g2)


k13 [( oI/IO)(Af)1/2 1 1
___ 1/2
k ,2 2 X o2guAt c Ma
14 -R 8 x B(T) 2 12 R in 2 o


k 82 in 2 Xo2(g+g2)c Mal/2 AtJ
15 ,I 8j( B(T)2 42 R in 2 o'

k atomic absorption coefficient at9 other than ),, cm.-i

L flame diameter, cm.

M amplification factor of photodetector, no units

Ma atomic weight, atomic mass units

Mf effective molecular weight of foreign species, atomic mass
units


MNaCl
N

n

N

Nf
Ni


molecular weight of NaC1, atomic mass units

total atomic concentration of species of interest, atom/cm.3

entrance optics factor, no units

ground state concentration of species of interest, atoms/cm.3

concentration of foreign species, particles/cm.3

atomic concentration at intersection (self-absorption) point,
atoms/cm.3


Nm minimum detectable number of atoms/cm.3 of flame gases








Nmo

nTI
"298

P

Pa

PC

Pe

Pe


Pf

Pi

PT

Q

R

Rd

Rf

Ro

s

sa

sc

Sd

Sm

T

AT

Tf

Tm
m

To

V


ground state minimum detectable concentration, atoms/cm.3

moles present at temperature T and temperature 2980K.,
respectively

power, watts

pressure of species of interest, mm.

partial pressure of C1 in all forms, atm.

partial pressure of electrons in the flame, atm.

partial pressure of electrons due to ionization of flame gases,
atm.

pressure of foreign species, mm.

partial pressure of species i, atm.

total pressure of species of interest in all forms, atm.

flow rate of unburned gases, cm. /sec.

gas constant, 8.3 x 107 ergs/moleoK.

reciprocal linear dispersion, mp/cm.

reflectance of optics, no units

resistance of load resistor, ohms

spectral slit width, mi

spectral slit width due to aberrations, coma, etc., mp

sd + sa' mP

diffraction limited spectral slit width, mp

spectral slit width as determined by mechanical slit width, mp

flame temperature, OK.

root-mean-square temperature fluctuation, OK.

transmission factor of optics, no units

transmittance of flame, no units

temperature of load resistor

ionization energy of atom, electron volts








v 2( Vo) in no units
A D

W slit width, cm.

Wo optimum slit width, cm.

y 2 i ln 2 no units
AVD

ZH number of Holtsmark broadening collisions/sec. atom

ZL number of Lorentz broadening collisions/sec. atom
a ratio of emission line width to absorption line width, no units

co width of beam of radiation at dispersing element, cm.

P fraction to account for incomplete compound formation and
atomic losses due to ionization, no units

'i abundance of isotope i, no units

Sphotosensitivity factor, amps./watt

A a variable distance from the point 9 Vo, sec.-

d factor to account for line broadening other than Doppler
broadening at 9 = ) o, no units

SV factor to account for line broadening other than Doppler
broadening at 9 other than V) no units

6 atomization efficiency, no units

a statistical weight fraction, no units

S ratio of intensity of hyperfine component i to the sum of the
intensities of all other hyperfine components due to
nuclear spin, no units

correction factor to account for multiplicity of spectral
line, no units

slit function parameter, no units

X wavelength setting of monochromator, cm.

Xo wavelength at line center, cm.

Q frequency, sec.-1

Qo frequency at line center, sec.-
o frequency at line center, sec.








V ratio of root-mean-square fluctuation in the background
intensity to background intensity, sec.1/2

t 3.14, no units

/0 correction factor to account for hyperfine structure of source
and absorption line, no units

cross section for Holtsmark broadening, cm.2

0l cross section for Lorentz broadening, cm.2

T lifetime of an excited state, seconds

0 flow rate of solution, cm.3/min.
1/2
X AIb/I, sec.

P --e/Ie, sec.1/2

We vibrational constant of molecule, cm.-

wi TiLk, no units















I. INTRODUCTION


The selection of optimum conditions is a problem which confronts

every analyst. To the analyst using atomic emission and atomic absorp-

tion flame spectrometry this problem is particularly vexing. Trial-

and-error methods of choosing optimum conditions are tedious and often

misleading because of the large number of variables and their inter-

dependence, and, until the present time, no adequate treatment has been

given describing the quantitative relationship of experimental factors

to the sensitivity of analysis. The work reported here was begun as an

attempt to take a more systematic and quantitative approach to the

question of optimum conditions for flame spectrometric analysis. How-

ever, the theoretical principles and expressions developed in this

paper have wider applicability than the prediction of optimum conditions

and, in fact, should provide the means for a quantitative discussion of

many of the phenomena and interference so frequently encountered in

flame spectrometry.

It is hoped that two major purposes will be served by this study:

first, to demonstrate that the influence of experimental variables on

the measured signal in atomic emission and atomic absorption flame

spectrometry can be treated quantitatively in a relatively simple

manner, and, second, that from this treatment optimum conditions of

analysis can be selected. It should be clear that if these purposes

are achieved this study will provide information of aid in routine

analysis as well as in theoretical considerations.










The approach taken in this study is to write expressions for the

signal and the noise as functions of the experimental parameters for

atomic emission and atomic absorption flame spectroscopy. The limit of

detectability is then defined as the atomic concentration of the species

of interest which gives a signal-to-noise ratio equal to two. The

resulting expression for the limit of detectability makes possible the

estimation of the effect of various experimental parameters on the sensi-

tivity of analysis and also allows a direct calculation of the limit of

detectability which may be expected. The selection of optimum condi-

tions for any concentration range of the species of interest is dis-

cussed on the basis of a maximum signal-to-noise ratio. The effects of

compound formation and ionization of the species of interest and self-

absorption of radiation are considered. Using the derived expressions

the calculation of optimum conditions is carried out for the atomic

emission flame spectrometric analysis of sodium, and the results of the

theory are compared with experimental measurements.















II. CALCULATION OF THE LIMIT OF DETECTABILITY FOR
ATOMIC EMISSION FLAME SPECTROMETRY

Introduction


The system under consideration in the development of this theory

consists of a flame source, a lens or mirror for focusing the emitted

radiation on the monochromator entrance slit, a monochromator with

accompanying optics and slits, a photodetector, amplifier, and readout

(recorder, meter, etc.). For the purposes of the theory, it is assumed

that the source is aligned for maximum intensity and that the slit is

fully and uniformly illuminated. The following discussion will apply to

either flames produced using total consumption or chamber type atomizer-

burners. However, when using atomizer-burners, flames are far from

homogeneous, and so it will be assumed that the brightest part of the

outer cone of the flame is always selected for viewing. For purposes

of simplicity the dimensions of the entrance and exit slits are assumed

to be the same, although, where the two slits are separately adjustable,

a small gain in sensitivity may result by having the exit slit slightly

larger than the entrance slit. The width and height of the slits are,

respectively, W and H. The effective aperture of the monochromator is
2
A, cm. and the focal length of the collimator lens or mirror is

F, cm. The term Tf is the transmission factor of the spectrometric

system (includes entrance optics as well as monochromator optics) as

determinedby absorption and reflection losses at all optical surfaces.









The spectral slit width of the monochromator is denoted s and

is given (6,8,29) approximately by


s = sm + Sd + sa sm + Sc [11


where sm is the spectral slit width of the monochromator when using wide

slits, sd is the diffraction limited spectral slit width of the mono-

chromator assuming focusing aberrations are negligible, sa is the

spectral slit width of the monochromator due to coma, aberrations,

imperfect optics, mismatch of slit curvature, etc., which results in a

circle of.confusion of constant size at the exit slit, and sc sd + sa.

In this paper the spectral slit width will always have units of mp. The

terms sm in mp is given (8,29) by


sm = (1/DF)W = RdW, [2]


where D is the angular dispersion of the monochromator in radians per

mp, and Rd is the reciprocal linear dispersion of the monochromator in

mp per cm. (i.e., A. per mm.). The term sd, in mp, is given (8,29) by


sd (1/D)X1/c FRdX/o [3]


where X is the wavelength setting of the monochromator in cm., and 06

is the width of the beam of radiation in cm. at the dispersing element.

The term sa, in mp units, is a constant characteristic of the mono-

chromator.

Equation [1] is not strictly correct because the above effects

are not strictly additive (6). However, equation [1] will give the

correct value of the spectral slit width when any one of the terms

predominates. If the above effects are of the same magnitude and are









independent, then the spectral slit width, s, will be given more

accurately by Pythagorean addition, i.e., s2 = Sm2 + Sc2. In this

section the additive relationship for s, given in equation [1], will be

used.

In addition to the above requirements concerning the entrance

optics and the monochromator, the following requirements will be made

for the system under consideration. It will be assumed that the detector

intercepts all the radiation passing through the exit slit and that the

amplifier-readout system is well regulated so that the limiting noise is

a result of detector and flame noise. The above assumptions and require-

ments regarding the instrumental system to be used are generally valid

for any good commercial flame spectrometer; and, therefore, the following

discussion should be directly applicable to many experimental systems.



Derivation of Equations


The minimum detectable concentration, N will be defined as that

concentration which produces an average output current anodicc current

of phototube) im, in amperes, such that'


im = 2 A [4]


where AiT is the root-mean-square fluctuation in the background anodic

current of the phototube. The value of i is given (20) by


im = YTfLmWH(A/F2)n [51


where y is the photosensitivity factor, i.e., the current in amperes

produced at the anode of the detector for each watt of radiant power

incident upon the photocathode, and Im is the total intensity (integrated









intensity) of the spectral line (in watts/cm.2 ster.) produced by the

minimum detectable concentration. (More correctly, Im should be called

the steradiancy (32) of the spectral line, but the more general term

"intensity" is used throughout with the units carefully specified.)

The term A/F2 is the number of steradians viewed by the spectro-

metric system as long as the effective aperture is filled with radiation.

This is true whether the source is viewed directly or a lens or mirror

is used to focus a selected portion of the flame on the slit. The para-

meter n is the number of solid angles of value A/F2 which are gathered

into a single solid angle by means of a suitable arrangement of entrance

optics. As has been pointed out by Gilbert (17), it is possible by the

use of a suitable system of mirrors to increase the intensity incident

upon the spectrometric system. Thus, if a mirror is suitable placed so

as to focus an additional image of the source on the slit, then the total

incident intensity is increased by n = 1 + RfTm where Rf is the reflec-

tance of the mirror and T is the transmittance of the flame for the
m
particular spectral line in concern. Using the spectrometric system

which is described at the beginning of this section, a single viewing

(n 1) of the flame is assumed. Once a theory is worked out for a

single viewing of the flame source, it is easy to evaluate n for any

number of viewings by proper consideration of the geometry of the

entrance optics. Gilbert (17) has considered several possible optical

arrangements for increasing the light gathering power.

The parameter V accounts for the position of the spectral line

with respect to the slit function distribution curve. For equal

entrance and exit slits, the distribution curve is triangular, and the

slit function parameter is given (8) by








1 [61
s

where X is the wavelength setting of the monochromator, and ko is the

wavelength of the line center. For monochromator wavelengths greater

than Xo+ s and less than Xo s, fCis zero. If the monochromator has

sufficient resolution to isolate a single, sharp spectral line, then the

value of ( can be made equal to unity by adjusting the monochromator wave-

length, X, to the peak wavelength of the spectral line, X0.

The integrated intensity Im in units of watts/cm.2 ster. is given

(2) by the well known equation


Im (10-7/4[)NmLh]oAt[gu/B(T)]eEu/kT [7


where Nm is the minimum detectable number of atoms/cm.3 of flame gases,

h is Planck's constant in ergs/sec., the 10-7 is necessary to convert

from ergs/sec. to watts, 0o is the frequency of maximum intensity of

the spectral line in sec.-1, At is the transition probability in sec.-l

for the transition from the upper state u to a lower state (usually the

ground state of the atom, designated o), Eu is the energy of the upper

state, k is the Boltzmann constant in units consistent with Eu, T is

the absolute temperature, and L is the average diameter in cm. of the

flame region being focused on the monochromator entrance slit.

The statistical weight of the upper state is gu, and the partition

function over all states is B(T), which is defined by


B(T) go + gleE/kT + g2-E2/kT.... e-Ei/kT [8



where the summation is carried out over all states. However, only in

cases in which there are excited states within 1.5 to 2 e.v. of the









ground state (e.g., Cr at 30000K in which B(T) is about 9 per cent

greater than go) does B(T) differ significantly from go and so in most

cases the simplificationB(T) = go can be made with negligible error.

More correctly, Nm in equation [7] should be replaced by Nm, the ground

state atomic concentration at the limit of detectability. The minimum
o 0
detectable concentration, Nm, is related to Nm by the expression N =

Nmgo/B(T), where, as discussed above, B(T)Sgo for most cases in flame

spectrometry.

As long as the spectral line is single, sharp, and isolated, the

value of Im is given by equation [7]. If radiation from more 'than one

line of the same element is passed by the spectroscopic system, then Im-

must contain additional terms for each line similar to the right hand

term of equation [3]. In this case im will be given by an equation

similar to equation [5], namely,


im TfWH(A/F2)n `djIjj [9]


where the summation is over all spectral components passed by the exit

slit. Since all components will not be passed centrally through the

exit slit, qj, the slit function for each spectral component passed by

the exit slit, will be less than unity for each line. The photosensi-

tivity factor, jj, for each spectral line passed by the exit slit will,

however, be nearly the same as the photosensitivity factor for the

wavelength at which the monochromator is set as long as the change in '

with wavelength is small over the spectral slit width of the instrument.

In any event it would be a relatively simple matter to determine the

value of (j for each spectral line passed by the exit slit by means of


~_~









the manufacturer's spectral response curves for the photomultiplier

tubes being used.

The evaluation of fj for each spectral component can be performed

by equation [6]. The greatest problem in evaluating the summation in

equation [9] is in determination of the number of spectral components

passed by the exit slit, i.e., as the slit width becomes wider more

spectral lines will be passed, especially for such elements as the

transition metals and the rare earth metals which have complex spectra.

A special problem results if spectral lines have wavelengths near the

extremes of the slit distribution.

If the spectral line is so broad that the intensity read by the

instrument is not the total line intensity, then additional factors must

be included in equation [9]. This problem has been considered by

Winefordner (41) and Brodersen (9). However, most spectral lines pro-

duced by excitation of atoms in flames are quite narrow compared with

the spectral slit width of most monochromators, and so additional

correction factors are rarely required.

It is not possible to consider in a general way the case in which

the spectral line is not resolved from lines of other elements because

this would require a detailed knowledge of the matrix in which the

analysis is to be performed. In the following discussion, the spectral

line will be considered single, sharp, and isolated, and the mono-

chromator wavelength will be assumed to be adjusted to the line center

(i.e., K,- 1). In this way the optimum slit width can be determined

by minimization of the limit of detection equation, Im will be given by

equation [7], and im will be given by 'im TfImWHA/F2 (t- 1 and n = 1).

As noted in the above discussion, there are cases in which the assumption









of a single, sharp, and isolated line is not valid. However, it should

be pointed out that even if in a particular case a line does not meet

the criteria of single, sharp, and isolated, it is still possible to

obtain exact results by substitution of the proper values into the

summation of equation [9]. More energetic sourcessuch as arcs, sparks,

etc., certainly do not meet the above criteria.

The root-mean-square fluctuation current, AiT, is a result of the

root-mean-square fluctuation current due to noise in the flame source,

Aic, and the root-mean-square fluctuation current due to noise in the

phototube, Aip. The flame noise is primarily due to fluctuation in the

intensity of the continuous background radiation, and the photodetector

noise is due to the shot effect and thermal noise. The two noise signals

add quadratically (19) so that AiT is given by


Ai (Ap2 + rc2)1/2 [10]


The value of Aip is given (35) by

1/2 [
ip (2ecAfBM(id + ic) + 4kTof/Ro) [


where ec is the electronic charge in coulombs, B is a constant approxi-

mately equal to 1 + 1/G + 1/G2 + 1/G3 + .... where G is the gain per

stage of the photomultiplier tube, M is the total amplification factor
-1
of the phototube, and Af is the frequency response bandwidth in sec.-

of the amplifier-readout circuit. Both flame noise and phototube noise

are assumed to be equally distributed over Af, i.e., the noise is

assumed to be white (19). The dark current in amperes at the anode,

produced by thermionic emission, is id, and the signal in amperes pro-

duced at the anode due to the incident intensity of continuous radiation









from the flame is ic. The temperature of the load resistor in OK is To,

and the load resistor of the detector circuit has a resistance of Ro

ohms.

The value of ic in amperes is given (see Appendix A) by


ic yTfIcWH(A/F2)s [12]


where Ic is the intensity per unit wavelength interval of the continuum,

i.e., Ic has units of watts/cm.2 ster. mp (called steradiancy per unit

wavelength interval). The value of Aic can also be shown to be given

(see Appendix A) by


Aic ~TfAIcWH(A/F2)s(f)12 [ 13]


where AIc is the root-mean-square fluctuation in the intensity (or

steradiancy) per unit wavelength interval of the continuum.

Combining several of the previous equations and solving for Nm,

one may write
7 Eu/kT
8*10 B(T)e
Nm h'TfWH(A/F2)V1guAtL iT [14]


Theequation for Ai can be somewhat simplified because the thermal

noise is generally (35) negligible when compared to the shot effect

for most systems and so equation [11] becomes

r 1/2
Aip = 2ecBMf (id + ic)] 1/2. 15]


Substituting for Aip, ic and Aic and collecting terms, the expression

for the minimum detectable concentration is obtained








Nm = [16]


3.8 x 1034B(T)eEu/kT(2ecBMcf [id + YTfIcWH(A/F2)s] + [CTf IcWH(A/F2) s 2f)1/2
9oguA LtTfWHA/F2


in units of atoms/cm.3 of flame gases. The constant 3.8 x 1034 results

when 8n x 107 is divided by Planck's constant.

If Nm in equation [16] is differentiated with respect to W and

minimized, then the condition for optimum slit width in cm., Wo, is found

to be satisfied when


[rT H(A/F IcRd 2Wo4 + [YTfH(A/F2)cI scRdWo3

ecBMYTfH(A/F2)IcscWo 2ecBMid = 0 [17]


By evaluating all factors in equation [17] for the particular experi-

mental setup in concern, Wo can be found. (Use of a graphical method is

the simplest means.) If the flame background is low at the wavelength

setting of the monochromator, then sm is much larger than s and s =

sm = RdW. For this case W is given by an expression considerably

simpler than equation [17], namely,

.2ecBMid 1/4
Wo ( [TfTcH (A/F2)Rd2 [18]


where equation [18] can be found directly from equation [17] by assuming

s is very small so that the second and third terms become negligible.

As Ic and AIc increase (AIc increases approximately proportionally with

Ic (17)), the last term in equation [17] becomes negligible when com-

pared to the other terms. Therefore, at large values of Ic, the

optimum slit width, Wo, decreases indefinitely and approaches zero.









The use of concentrations in atoms per cm.3 of flame gases would

have little meaning to most chemists. It would be more convenient to

use concentrations in moles per liter of solution introduced into the

given flame. If the concentration of sample is C moles per liter and

if the sample is introduced into a given flame at a rate of 0 cm.3/min.

or 0/60 cm.3/sec., than 0/60 x 1000 liters or CO/60 x 1000 moles of

sample are introduced each second. The number of atoms per second intro-

duced into the flame is then C06 x 1023/60 x 1000. If Q is the flow rate

of unburned gases in cm.3/sec. introduced at room temperature and one

atmosphere pressure, then C06 x 1023/60 x 103 Q is the number of atoms

per cm.3 of flame gases.

This expression assumes thorough mixing of the sample with the

flame gases, no entrainment of atmosphere, no expansion of burnt gases,

100 per cent efficiency of sample introduction (42), and complete dis-

sociation of the salt crystals into atoms. With a total consumption

atomizer-burner nearly complete mixing of sample with flame gases occurs

in the outer cone of the flame. No detailed equations can be given to

consider the extent of atmosphere entrainment because it depends on the

exact experimental arrangement. The expansion of burnt gases has been

accounted for by Alkemade (1) by use of an expansion factor ef.

Winefordner, Mansfield,and Vickers (42) have accounted for incomplete

dispersion of sample solution into droplets and incomplete solvent

evaporation by use of a sample introduction efficiency factor e. A

factor P will be used to account for incomplete dissociation of salt

crystals into atoms and atomic losses due to ionization and compound

formation with flame product gases. Therefore, using ef, e,and P in









the previous equation, the flame gas concentration N in atoms per cm.3

of flame gases can be converted to solution concentration C in moles

per liter by

6 x 104. efQN
C 6 x 1023. [19]


The term ef has been evaluated by Alkemade (1) and is given by


ef = nTT/n298 298 [20]


where T is the flame temperature in oK., nT is the number of moles of

combustion products at temperature T and n298 is the number of moles of

species at room temperature, which is taken as 2980K. in this case. Thus

the final form of the conversion equation is given by


C = 3.3 x 10-22 nTQTNer. 21
moles/liter. [21]
n298 6


In the above equation, the values of nT and n298 must include not

only the moles of flame gas products due to the fuel combining with

oxygen but also the moles due to the vaporization of the solvent intro-

duced. Equation [21] can be used to calculate the minimum detectable

solution concentration of a given atomic species once the minimum de-

tectable concentration of atoms in the flame has been calculated.



Discussion and Calculations


Equation [16] for Nm is a general expression and should allow the

accurate calculation of limiting detectable concentrations for any









element, present in any given flame, and analyzed using any given

experimental setup, as long as accurate data are available to evaluate

the factors. In many cases good data will not be available, and this

will prevent accurate absolute values of N from being calculated. How-
m
ever, it should be stressed that the usefulness of equation [16] lies

not only in the calculation of detection limits but also in the pre-

diction of the effect of changing various experimental factors on the

limit of detection. In addition, if sufficiently accurate data are

available, then it should be possible to determine accurate transition

probabilities if accurate measurements of absolute intensities are made.

It should be noted that equation [16] for Nm is exact as long as

the spectral line is single, sharp, and isolated, and if this is not

the case, then the other spectral lines passed through the exit slit

must be corrected for in the manner discussed in the preceding portion

of the paper. Also the equations derived should apply whether single

stage phototubes or multistage phototubes (photomultipliers) are used

as detectors.

The use of equation [17] for calculating Wo and equation [16] for

calculating Nm is illustrated by the representative results in Table 1

for Na 5890 A. and Cd 2288 A. lines analyzed using the experimental

conditions and data listed at the end of Table 1. However, equation

[18] gives essentially the same value of Wo as equation [17] because

of the low flame background.

The values of F, A, Tf, H, and Rd listed in Table 1 were repre-

sentative values taken from the manufacturer's literature. Photo-

multiplier tube characteristics vary with individual tubes and for









accurate work id, M, and ( must be experimentally measured for the

particular detector. For the purpose of this paper the value of id

given by the manufacturer for a good 1P28 photomultiplier has been

taken. Because the amplification factor, M, is related to the anodic

photosensitivity, j, by a constant characteristic of the photocathodic

surface and because the anodic dark current id equals M times the

cathodic dark current, the value of N is independent of M and T.
m
However, characteristics of phototubes are generally reported in terms

of anodic rather than cathodic output, and therefore, anodic dark

current and anodic photosensitivity are used in the theory of this

manuscript. The values of M and Y given in the manufacturer's litera-

ture have been taken as typical. The values of Y at the wavelengths

of interest were corrected for variation in spectral response from the

manufacturer's response curves. The gain, G, per stage of a photo-

multiplier tube is of the order of 4, and so the value of B is taken as

1.3.

The absolute background intensities, Ic for oxyhydrogen and

oxyacetylene flames in units of watts/cm.2 ster. mp have been taken

from the plots of Ic versus wavelength given by Gilbert (17). Gilbert

also gives similar plots for air-hydrogen and air-acetylene flames.

The value of AIc is given by cI,, where y is the ratio of the root-

mean-square fluctuation in the background intensity to the value of the

background intensity. The value of f depends upon the temperature

fluctuations, irregularities of atomization at the spray tip, mechanical

turbulence in the spray and the burning gases, and turbulence due to

friction and mixing with the ambient air (17,20). Of these factors only
















od
a


SS


F -

O

-4 0



OO









U 0

0 0
W HO




prz




0 C
0 Q





a0 a
0



c eO M






H4

H
co 0






a
So







> C
E-4<
B1


0
Q)





0 0 *0



0
, I
oe e C








0 CM4
U U 0

0 u









S 0
0







0






u
p z
4J




Q)o





.0 C4
0









4-)
a










n Z


0'1 0




















-"H
CO 0
O




0
4J







U,
0
0 -



o a



0











-4 3





o o
rI 4


0


r-4
CO

C4
cu




0








Le



-4
-I









0









0
0

a
-4

co
-4






















\o
0
O
0








0
-4




-4




0







4-4





o
0
zo
*


.N
-41


0
-r4
41
O




0




:3
Cd





















r-44
-I



0
a
bo





0l


0
0
0

-4
,O



:J
-4


u



r--
-I
0

















.1
0
o5

0
0f



i-


0
o






0










-4
M

0

a-
a0













-4
a41



rl0

il
0
*U
1-

M


00



















4J
CO,































N
a-


0



0u
OI
co
0




0O

0
u






-4





II
a

0




















C--'
II











0
6





0








I
fto


*Il

a *
a)
L0
0 -4
01
o41
II C0






I0 :




C 4- 1
0



ca
.>
Q1)
-,C



cc4






-4 0
0

4-)
u --*
c0

CO



0 0c
-4
H 0
U II


a





p .
CO


r4
0
(0




411



0
to.





0
00 (d







a m
a

o ico



o
o
0 II






-1

a


*

00

0

a-I C5
i-0 01
0-cd


> r~cc
40
a 0


-14 4-4

n ) X
S- 0











Q) 00
o in
-H* ai





I-ca4
0 )







C a

r4-
0 ca



CN cc



Cu C)
00



0a 41 4o
o 41










p3





0 0
cc




0










-40 0
0 a










c 5i
C C











Tl C
Ue 1
i






0 4J

a



.0 0- rl







me


S0
cda

00




&4
oa

0
oo

co
u *

4 ^
cc
4 0 M







0

0 a






-4 M .
0 u

0 0 0
0 () 0
Fj 04(





OH
ca
s-4


4 60




0 .-

,* 0 0
-4a a

On .o
(3 *




4 -0
0








0 0 pH
0 0 cc


M 00



0 0
Sed








4 -rt *

co 4
c-- -0 0
c'




5 -1





co in
*d c* I



O 0

0 *
M 0





CM00


C)
H
ul


a


II
0




in
o


0
0





0c
a


o
o

0
0











Co
0


4,a

-4
4

-4













0
aO









4o
0












4J
vu
a,0
-d
rl
01
03










ai
0
0
-4











01
0)




i--
-4
4-







0
0


co
CM




0
4-i

0


0
a






i4
d






II






O





H--1
II







II















-~4
00
*










v-40
I





o co
II




0 *
IM


















.4
T- 0
U II








00a
O .
0
c-




SII






a
0 O4

0 II














O -H









the fluctuation due to temperature variation can be readily represented

by an expression. The background intensity of the continuum varies

approximately as exp (-hV/kT) and so an approximate value of can be

found by differentiating Ic with respect to T to find the relationship

between I and AT, the root-mean-square temperature fluctuation. If this

is done, is found (1) to be given by (hV/k) (FT/T2), which will be

smaller than the true due to neglect of the other factors mentioned

above. A Beckman flame in good adjustment (at 30000K. and 5000 A.)

results in a value of of approximately 0.005. If only temperature

variation is considered, this value of will result from a temperature

fluctuation of about 20K. The use of a sheathed burner as described by

Gilbert (18) allows even better stability. Experimental values of

can be measured by finding the magnitude of the peak-to-peak noise as

compared to the value of the background signal and multiplying by

42 /4 to convert to r.m.s. noise (19).

By the use of equation [21], the Nm values in Table 1 can be con-

verted to more meaningful solution concentrations in moles/liter. For

a typical oxyacetylene flame with a temperature of 28500K. at 1 cm.

above the inner cone (43), when an aqueous solution is introduced into

the flame via a total-consumption atomizer-burner, at a flow rate of

2 cm.3/min., a concentration of 1.8 x 106 atoms of Na per cm.3 of flame

gases was found to be equivalent to a solution concentration of

1.6 x 10-4 p.p.m. This value agrees quite well with the value given by

Gilbert (17).

In this calculation the flow rates of C2H2 and 02 were, respec-

tively, taken as 2500 cm.3/min. and 3750 cm.3/min. This would result








in a value of Q of 125 cm.3/sec. if allowance is made for the sample

vapor, assuming that the sample vapor is not appreciably dissociated (46).

The data for nT and n298 were taken from the thesis by Zaer (46). The

ratio nT/n298 is approximately 1.2. The efficiency of atomization was

taken as approximately 0.5 according to data by Winefordner, Mansfield,

and Vickers (42).

The value of P was determined primarily by ionization of sodium

atoms, although the formation of NaOH in the flame (16,24) may cause P

to be slightly less than by ionization alone. An approximate value of

P-was determined in the following way. The ionization constant for the

process Na -- Na+ + e- was calculated to be 1.01 x 10-7 atm. at

28500K. from the expression given by Mavrodineanu and Boiteux (30).

The partial pressure of electrons due only to the flame gases was taken

as approximately 4 x 10-9 atm. (15) in the outer cone of a C2H2/02 flame

at 28500K. This partial pressure of free electrons corresponds to 1010

electrons per cm.3 of flame gases assuming the flame gas solution of

free electrons is an ideal gas. Using the above data, the degree of

ionization was found to be 0.96 and, therefore, P was found to be 0.04.

The degree of ionization calculated agrees quite well with values cal-

culated by Gaydon and Wolfhard (15) and Foster (12) for the outer cone

of acetylene flames.

The above calculation did not account for additional dilution

of the analyte vapor due to entrainment of ambient air. Except under

accurately controlled conditions, this additional effect is not deter-

minable. If this effect is accurately known, then Q and nT/n298 should

be correspondingly corrected. However, this is unnecessary unless the









most accurate possible results are needed, such as when the described

theory is applied to the calculation of transition probabilities for

unknown cases. If this is done, it is best to use well defined,

sheathed., homogeneous flames in order that Nm is accurately known.

In Table 2 the effect of flame temperature on Wo and Nm is given

for the resonance lines of Na and Cd when analyzed by a Beckman DU mono-

chromator IP28 photomultiplier detector combination. The resonance

lines of Na and Cd differ quite greatly in wavelength and in excitation

energy and were especially chosen to illustrate the influence of

excitation energy on the limit of detectability as a function of several

flame temperatures. No single flame type can be used to cover the broad

range of flame temperatures listed in Table 2 (see item 2 at end of

table), and so the flame type is not specified. However, a stoichio-

metric air-acetylene flame has approximately a temperature of 20000K.,

a stoichiometric oxyacetylene flame has approximately a temperature of

30000K., and an oxycyanogen flame has approximately a temperature of

40000K. when aqueous solutions are introduced from total consumption

atomizer-burners at the rate of about 1 cm.3/min. The values of Ic

listed in Table 2 were considered typical and should not be considered

as accurate values for the above flames.

As can be noted from Table 2, an increase in flame temperature

results in an improvement in limit of detectability and a decrease in

optimum slit width. With elements having their resonance lines in the

u.v., an increase in flame temperature gives a great increase in sensi-

tivity as a result of the exponential factor in equation [16]. How-

ever, for elements with their resonance lines in the visible the increase










1-
On 0 I 0
0 0 o i- n $
-4 ,-4 :j u u),4 0
4-1 Q> (U) a :3

00 1 M 0 0o> C
a W up u o co u-4

-M < C ud l E-4 c

-4 0 O C00 00 P

0 n 0 0 0-00 0 *4
0- .a O 0 ) 4 40
H 0 oO 0 r- 0 .
E-4 0 -4 0 Q r


HZ 0O- 0 0 0 ,
ZOO 0> 4 0
0 C 4 4 4-

- > 00 0
>-3 E0l l 4l 44-4
-- 4 04 >iO 0 0 0

p-3 [.0 U .* 0 0 4
HOM 0
S E- 044 ( 00 4 Q 4 r -400
E 0 00 4- 0 40 4 0
O-4 ,- 0 l "4 l m)J -4) a
H -4 OOHO U.r 4p 4a
w0 o -4 0 4l O l 6
up y oo r 4) p a u "l a) 4 Q l


[-.U 0 (0 0 ,- 0 4J-4
0 41 4 q
OP14 0 4 r00
0 0 I 4 0 a) a 4
0H 0 0 00 0 *0 O 30 i
(1 0 00 0 O -- *-4


l 00 0 -0 > lt 4 C
H t H Cu 0OU-) u -4r 0 -
Q<) 0 l O 0 X4 -,l cU
SH C *- 4m) 41 t 00 0 (1
Z Z-i o N XN^ 41 0
>% 0's C 0u *41 CU z
cu po -a
C ca : M 0 l Ml X )
20 o 0 -o40 4 l0 4 41
0.0Q 4.Ha
H4JO O 0 0 C

-) \< 17 3 "m CNO i 0) gl f r
P. i 00 0 0 C O
H 0 p l'-4 0 0
w '-4ue4- ") 00 l
uO 0 % 0 C0 O .4 4J 0 0
4 r- 0 3 U0 0 r 0 *4

x x U 0 H 40

.E-D (U) 41 4 C0 e
H 0 C :.% C. l



O CIJ-r:. 0 O 3c 0 E
'0 *2 o > *6 o0
E-4 W c4 -L4 O 4J .4 *i0 u co
p S a CO 44 U )
(. 4 -',4 *O 0
SE- 0 ( 0 C
)E-4 0 o C1 r4 0 a 3 4
0o ZM r r=: 0 No
Cl.1 U) M M 4J PQ Ca u
Z >*- 4 C 0
40 1* 41 1 0 4 a0 00 0
w- Ca 1 U 3 C 4




0 0 * CE-I ) C u-
/3 U 4 a e bO -4 3 0b14
0-0 0 B -.-4 -Li
O W -0,4 O kD 0 a : 0) 404 ()




0 0 aw 44 0m o *4
> > HO0M 0 w 0A


w E- >

0 O- I *- E4 W4w 0 a)
13 4 4 () 04 q 0 (O


VI 0 C : 0) 0 mU 01 3
E, ( e *O uo
0 n c4 rA u o I >
0 o () r-4 ( O 4 o u 40 t0 Ht
41 a V 4 0 pQ 0 0 a 0 -
44 z< u 0- o J O









in sensitivity, i.e., decrease in value of Nm, with flame temperature

is not as great as might be expected. This neglects any adverse effects

due to ionization and is due only to the increase in Ic and aIc asso-

ciated with an increase in flame temperature and a smaller increase in

the exponential factor of equation [16]. Therefore, for elements with

spectral lines in the visible, no significant advantage may result by the

use of higher temperature flames, e.g., oxycyanogen or fluorine-hydrogen,

rather than the usual flames of oxyhydrogen or oxyacetylene (or even air-

acetylene), and in some cases an exceptionally high background may result

in a decreased sensitivity. The data used to obtain the results given

in Table 2 are listed at the end of the table.

The shape of the Nm versus slit width, W, curve for any particular

experimental setup and spectral line should be of special interest to

the analyst. If this curve has a broad flat minimum, then the exact

setting of the slit width for optimum conditions is not critical.

Because the shape of the Nm versus W function is independent of the

spectral line, it is not necessary to evaluate all factors in equation

[16] at a variety of slit widths, but rather it is possible to combine

all factors except 1/W outside the square root sign into a constant

factor designated Kc. This constant Kc is a function of the particular

line, flame and instrumental setup but is independent of W. Therefore,

by replacing s by RdW and evaluating all universal constants, equation

[16] can be rewritten as


Nm = [22]


(Kc/W)(5 x 10-19M[id + TfIcH(A/F2)RdW2] + [YTf1cH(A/F2)Rd]2W4)1/2









where all factors within the square root must be evaluated for the

particular experimental conditions. It is evident from the above

equation that the shape of Nm as a function of W is not dependent on

the magnitude of K In Table 3 values of Nm have been calculated from

equation [22] for various W values, and, as expected, the minimum value

of N occurs at Wo.

As can be seen from the results in Table 3, Nm varies approximately

linearly with W for large W values and 1/W for small W values. The

particular experimental conditions for which the calculations were made

are given at the end of the table. For the experimental conditions

listed in Table 3, the Nm versus W curve has quite a flat minimum. In

fact the value of W can deviate from Wo by a factor of 2, and Nm in-

creases by a factor of no more than 1.4.

The above discussion regarding equation [22] is valid only if the

spectral line is resolved as a single, sharp line. If two or more

spectral lines are collected by the exit slit, then equation [22] must

be modified accordingly, as previously discussed. In the latter case,

a plot of Nm versus W may result in more than one minimum of Nm for

certain optimal wavelength settings. For example, with hydrogen flames

and prism monochromators, the most sensitive spectral line of elements

such as Fe, Co, Ni, Ru, Rh, and Cr are not fully separated. Even the

best prism monochromator with the narrowest useable slits may not be

able to isolate the sensitive lines of the rare earths.

Several other cases will be given which demonstrate the effect of

various factors on the optimum slit width. A small value of Wo

(Wo = 0.002 cm.) results when using equation [18] for the following











VARIATION OF


TABLE 3

Nm WITH W FOR A PRISM MONOCHROMATOR AT 420 mp WHEN
USING AN OXYHYDROGEN FLAME*


W, cm. Nm


0.001

0.005

0.008

0.010

0.012

0.015 (Wo)

0.020

0.030

0.050

0.100


7.1 x 10-8K

1.5 x 10-8Kc

1.0 x 10-8Kc

0.90 x 10-8Kc

0.82 x 10-8Kc

0.78 x 10-8Kc

0.79 x 10-8Kc

0.91 x 10-8K

1.3 x 10-8Kc

2.6 x 10-8K


*The above calculations were performed using equation [22] and the
following data: id = 10-8 amp. (for a good 1P28 photomulti lier tube),
= 104 amp./watt, M = 106, Tf = 0.5, H = 1 cm., A = 25 cm. F =
50 cm., Rd = 100 2m/cm., Af 1 sec.-1, and AIc = 5 x 10-11 watts/cm.2
ster. mp sec.

Using the above data and equation [18], Wo is 0.015 cm.









conditions: id = 10-8 amperes (a good 1P28), W = 104 (a good 1P28 at

400 mp), B = 1.3, M = 106, Tf = 0.5, Rd 100 mp/cm. (for a prism mono-

chromator at 400 mp), AIc = 3 x 10-9 for an oxyacetylene flame at 400 mp,

and HA/F2 = 0.01. For this case the value of W is of the same order

as the practical (effective) resolving power expressed in terms of slit

width. In the Beckman DU monochromator, the practical resolving power

slit width (20) is about 0.003 cm. in the visible and 0.006 cm. at the

shortest useable wavelengths, and it is determined by the combination of

diffraction, coma, aberrations, slit curvature mismatch, and optical

imperfections. Therefore, for this particular case, the limit of

detectability will be considerably lowered if the slit width is made

smaller than the effective resolving power slit width although no gain

in resolution will occur. If in the above case the values of Ic and

AIc are increased by 103 corresponding to a brilliant background in a

plasma flame, Wo is approximately 0.00007 cm., which is considerably

less than the effective resolving power slit width.

A case in which a wide slit width value of Wo results would be

the case in which id 3 x 10-8 (maximal specification for a 16PMI

photomultiplier tube), Y= 102 (at 630 mp), B = 1.3, M = 105, Tf = 0.2

(for a grating monochromator), Rd = 10 mp/cm., HA/F2 = 0.005, and AIc =

3 x 10-12 watts/cm.2 ster. mp sec.-1/2 (a good stable H2/air flame at

630 mp). For this case Wo = 3.4 cm., which is a very wide slit.

Assuming no lines are present other than the line of interest, it is

interesting to find how much sensitivity is lost if a narrower slit is

used. From equation [22], Nm is 1.4 times greater at W = 1.6 cm. than

at W = 3.4 cm. and is 4 times greater at W = 0.5 cm. than at W = 3.4 cm.






26


In this case the value of Wo found by using equation [18] is so large

that it is essentially the same as that given by equation [17].














III. CALCULATION OF THE LIMIT OF DETECTABILITY FOR
ATOMIC ABSORPTION FLAME SPECTROMETRY

Introduction


The system under consideration in the development of this theory

consists of a hollow cathode discharge tube (HCDT), or other resonance

lamp which emits sufficiently narrow lines, a flame into which the

sample is aspirated, a lens for focusing radiation from the source onto

the central part of the outer cone of the flame (which can be considered

to be approximately in thermal equilibrium), a second lens for focusing

the transmitted radiation onto the monochromator entrance slit, a mono-

chromator with accompanying optics and slits, a photodetector, amplifier,

and readout (recorder, meter, etc.). The HCDT emits a line of intensity

10 (in watts/cm.2 ster.), and the intensity of the transmitted radiation

is It (in the same units as 10). The transmitted radiation is focused

by the second lens onto the entrance slit of the monochromator so that

the slit is fully illuminated and the effective aperture of the spectro-

metric system is filled with radiation.

It is assumed that correction is made for thermally emitted and

fluorescent radiation of the same frequency as the incident radiation

so that It is the true transmitted intensity, i.e., it does not include

emission from the flame other than flame noise in the frequency interval,

af, over which the amplification-readout system responds. Also it is

assumed that correction is made for any radiation loss due to scattering

by droplets in the flame. Further it is assumed that the monochromator










is set on the line center, yo (the frequency of the absorption line

peak in sec.'1), and that only radiation of the line of interest enters

the spectrometric system.



Derivation of Equations


If the spectral line emitted by the source is much narrower than

the absorption lines in the flame, then 10 is related to It by Beer's

Law


A = log (l1O/t) = 0.43koL [23]


where Ao is the absorbance, ko is the atomic absorption coefficient in

cm."- at the line center for any atomic concentration, and L is the

length in cm. of flame gases through which the radiation is passed.

Justification for the assumption of a narrow source line is to be found

in the statements of Jones and Walsh (26) and Crosswhite (11) and in the

ability of the experimenter to adjust the source line width by changing

the operating current of the tube, or, if necessary, even by cooling the

tube. (However, see Appendix C for a further discussion of the effect

of line width.)

Thus it is seldom necessary to operate under conditions in which

the effect of Doppler, Lorentz, and Holtsmark line broadening is more

important in the source than in the flame. However, it may occur that

a line is intrinsically broad due to hyperfine structure, both from

nuclear spin effect and isotope shift. It may be shown (see Appendix D)

that the effect of hyperfine structure is to reduce the absorbance

below the expected value. Thus









A = log (IO/It) = 0.43pkoL [24]


where p is the correction factor to account for the decrease in

absorbance. The method of calculating p is indicated in Appendix D.

The absorbance will be said to be just detectable when the

difference between the detector output signal anodicc current of the

photodetector) i, in amperes, due to I1 and the detector output signal

it, in amperes, due to It equals twice the total root-mean-square noise

fluctuation. It is assumed in this theory that the noise is measured at

the wavelength of the line of interest with solvent aspirated into the

flame.

Thus the minimum detectable signal difference is given by


(io it)m = 2iT [25]


where AiT is the total root-mean-square noise fluctuation in the output

current of the photodetector in amperes, and the subscript m indicates

that the signal difference is that observed at the minimum detectable

concentration. The relationship between signal in amperes and intensity
2
in watts/cm.2 ster. has been derived in Appendix A. Thus it may be

shown that


i Y-TfWH(A/F2)Io [26]


and


it = TTfWH(A/F2)It [27]


where all the terms have been previously defined.









Substituting for i and it in equation [25] one obtains


YTfWH(A/F2)(1 It)m 2 [28]


However, from equation [24], It can be written in terms of 1 to obtain


YTfWH(A/F2)I(1 e- kmL) = 2A [29]


where ko is the atomic absorption coefficient at the line center at the

minimum detectable concentration. Expanding the exponential term and

noting that komL is small at the limit of detection, i.e., e-pkOmL

1 -pkmL, one may write


TfWH(A/F2)IokomL = 2ZiT [30]


The root-mean-square noise signal, AiT, may be attributed to

fluctuations originating in the photodetector, Aip, fluctuations in the

source, Ai, and fluctuations in the flame continue, Aic. The noise

signals add quadratically (18), and therefore, AiT is given by

-022 -- 2)1/2
AiT (Aip + o + i2 2 [31]


The value of Aip is given (35) by


6ip = [2BMecAf(id + i + i)]1 [32]


where B is a constant approximately equal to 1 + 1/G + 1/G + ...,

where G is the gain per stage of the photodetector, M is the total

amplification factor of the photodetector, ec is the electronic charge

in coulombs, id is the photodetector dark current in amperes, and ic is

the anodic current of the photodetector in amperes due to the continuous








emission of the flame. In any practical case in atomic absorption flame

spectrometry id and ic will be much smaller than 1i or, at least, can be

made much smaller at the limit of detectability. Hence one may write


Aip (2BMecAfi)/2 [33]


and substituting from equation [26] one obtains for 2i


i-p2 = 2BMecAfXTfWH(A/F2)Io [34]


The value of Ai2 is related to the fluctuation in the intensity

of incident radiation by


o2 = [YTfWH(A/F2)To]2Af [35]


where Al- is the fluctuation in the intensity of radiation from the

source in units of watts/cm.2 ster. sec.-/2 The sec.-1/2 unit arises

because the noise is a root-mean-square noise. The relationship between

the noise signal and the response bandwidth, Af, in sec.-1, has been

more fully discussed in Section II.

Similarly, Aic is given by


Aic = [TfWH(A/F )Acs ] Af [36]


where AIc is the fluctuation in the intensity of the continuum emitted

by the flame in watts/cm.2 ster. mp sec.-1/2, and s is the effective

spectral slit width (in mp) of the spectrometric system.

The spectral slit width is given by the expression


s (m2 + sc2 1/2 [37


where sm is the spectral slit width in mp as determined by the mechanical








slit width, and sc is the spectral slit width in mg at an infinitely

narrow mechanical slit width as determined by diffraction and by coma,

aberrations, mismatch of slit curvature, imperfection of optics, and other

factors characteristic of an imperfect monochromator. Expression [37]

differs from the one given for s in Section II since in the atomic

absorption case, the more exact equation for s can be used without overly

complicating the theory.

It will be convenient in calculating the noise terms to write the

fluctuation as fractions of the total intensity such that AIo XIo and

AIc Ic, where X and f are the appropriate fractions, and Ic is the

intensity of the continue emitted by the flame in units of watts/cm.2

ster. mp. Equations [35] and [36] may then be rewritten in the following

form:


A02 [YTfWH(A/F2)xIO]2Af [38]


Ai2 = [YTfWH(A/F2 )Ics ]2l f [39]


Substituting the appropriate values into equation [31] one obtains for

AiT
i--T = 2BMecIo 2 1 /2

Ai = [YTfWH(A/F2)]2 ( YTfWHA/F2 + (X0) + (cs)2 [40]

The value of kom, the atomic absorption coefficient at the line

center at the minimum detectable ground state concentration Nom, is

given (31) by


o 2 ln o2 gu NomAtf 0.037 o2guNomA [41]
km = ui, [41]
AV D ~8t8go go0VD


where A'D is the Doppler half width (in sec.- ) of the absorption line,









Xo is the wavelength (in cm.) of the line center, gu and go are the

statistical weights of the upper u, state and the lower (ground, o)

state, At is the transition probability in sec.-I for spontaneous

'emission, i.e., state u to state o, and S is a factor to account for

line broadening other than Doppler broadening and hyperfine structure

broadening (see Appendix D). Because analytically important atomic

absorption transitions usually involve the ground state, only transi-

tions to that state will be considered. The constants have been evalu-

ated in the right hand part of equation [41].

The number of ground state atoms per cm.3 of flame gases, N,

is related to the total number of atoms in all states per cm.3 of flame

gases, N, by the famous Boltzmann equation, namely No = Ngo/B(T) where

go is the statistical weight of the lowest (ground) state and B(T) is

the partition function of the atom, i.e., B(T) = ZgeEikT where

gi is the statistical weight of the state i, Ei is the energy of state i

above the ground state, and the summation is over-all states of the atom.

For the case of the minimum detectable concentration, Nom = Nmgo/B(T).

As previously discussed in Section II, B(T) is approximately given by go

except for a few atoms with low lying atomic states, e.g., Cr. In this

paper B(T) for the examples given is accurately expressed by go and so

Nm is essentially the same as N The above theory is still correct

for transitions occurring to levels higher than the ground state if go

is replaced by the statistical weight of the state involved.

Substituting for kom and AiT in equation [30], and solving for

Nm one obtains

53.5 AVDB(T) 2BMec 2 + ) 1/2 [
pNm LX, TWAT2) + 2Tf ( s'/I) AfguA [42]
LAo28guAti TyWH(A/F2 o









The above equation was derived assuming that the output photodetector

signal due to thermal emission, fluorescent emission of the resonance

line,and incident light scattering was either negligible or corrected

for. In Appendix E equations for Nm are derived for the case in which

compensation is not made for thermal emission for several detection

systems. The influence of fluorescent emission and light scattering are

also considered in Appendix E.

The optimum slit width, Wo, can be obtained by minimizing equation

[42] with respect to W, i.e., equating dNm/dW to zero and solving for

W. If s is given by s = (RdW)2 + Sc2 1/2 from equations [2] and [37],

and if sc is assumed constant with change in slit width, then Wo, in cm.,

is given by

( BMecIo 1/3
Wo = yTfH(A/F2)(qIcRd)2) [43]


The minimum detectable concentrations in atoms/cm.3 of flame

gases can be converted to solution concentrations in moles/liter by use

of the equation derived in Section II (equation [21]).



Discussion and Calculations


When one attempts to calculate minimum detectable concentrations

through the use of equations [42], [43], and [21], the need for further

measurements of basic experimental parameters of atomic absorption flame

spectrometry becomes apparent. One of the most important contributions

of these equations is that they point out those areas most in need of

further work. It is difficult to compare calculated limits of detectability

with measured limits of detectability unless accurate absolute values of









1, Ic, X, Y and the photomultiplier tube characteristics are

stated. Also, more information is needed on I values and on the degree

of compound dissociation and atomic ionization so that P may be calculated

with some accuracy. Although the calculation of limits of detectability

is at present hampered by this lack of information, equation [42] is still

valuable in that it allows one to calculate the effect of varying one or

more of the experimental parameters on the sensitivity of analysis, e.g.,

the effect of slit width or temperature on Nm.

Table 4 shows the results of calculations of the minimum detectable

concentrations for two favorable cases. When one chooses a good instru-

ment such as the Beckman DU or the Jarrell-Ash 500 mm. grating mono-

chromator, a 1P28 photomultiplier detector, and flame conditions such as

stoichiometric oxyhydrogen or oxyacetylene, then, assuming a reasonable

value for 10 and using the optimum slit width, all the terms under the

square root in equation [42] except X2 are negligible. It is assumed

that correction has been made for the output signal of the detector due

to thermal and fluorescent emission and light scattering. Thus for these

conditions, which are of experimental importance, the value of N de-

pends directly on X the fluctuation of the source intensity, and so

equation [42] can be reduced to


53.5AV DB(T) X /2
Nm = .pI[2guAt [44]



Equation [44] is in agreement with the expectation that over a

fairly broad range of experimental conditions, the sensitivity of atomic

absorption flame spectrometry would be independent of instrumental

parameters. For the case represented by equation [44] the limit of








TABLE 4

REPRESENTATIVE RESULTS IN ATOMIC ABSORPTION FLAME SPECTROMETRY FOR
LIMITS OF DETECTABILITY IN ATOMS PER CM.3 OF FLAME GASES
FOR Na and Cd IN TWO FLAMESa

Flame Temp.
OK.

Atomic Line 2700b 2850c

Na 5890 A.d 1.1 x 1010 5.6 x 109

Cd 2288 A.e 4.3 x 1010 2.2 x 1010


aX = 10-2, Af

bStoichiometric
2 cm.3/min., L


= 1.


H2/02 flame with aqueous solution flow rate of about
= 0.5 cm.


CStoichiometric C2H2/02 flame with aqueous solution flow rate of about
2 cm.3/min., L = 1 cm.


dAVD = 4.1 x 109 sec.-1 at 27000K., 4.2
B(T) g=^=2, g = 4; p = 0.97; Xo
1.3 x 108 sec.-l;u = 0.43.


x 109 sec.-1 at
= 5.89 x 10-5


eVD = 4.6 x 109 sec.-1 at 27000K., 4.7 x 109 sec.-1 at
B(T) go = gu = 3; ( 1; X = 2.29 x 10-5
1.7 x 108 sec.-1; u 0.43


28500K.;
cm.; A =


28500K.;
cm.; At









detectability could be greatly improved only by an increase in path

length (13) or an increase in the stability of the source. However,

from equation [42] it may be seen that as the stability of the source

is improved, other terms might become significant. The term (FIcs/Io)2

becomes important as the flame background-increases, e.g., with the

highly fuel rich flames frequently used with elements that form very

stable oxides, or as the spectral slit width increases, e.g., with

filter instruments. However, for the conditions listed under Table 4,

the minimum detectable concentration may be calculated from equation [44].

The values of the parameters given in Table 4 were found as

follows. The Doppler half width, AVD, can be obtained from the expression

given by Mitchell and Zemansky (31)


AVD 2 2Rn 2 T( ) 1/2
cMa

7.15 x 10-7 ~o0 ) [45]



where R is the gas constant in ergs/mole. oK., c is the speed of light

in cm./sec., T is the absolute temperature, and Ma is the atomic weight

in atomic mass units. Universal constants have been evaluated in the

right hand part of equation [45]. Values of 9 in terms of the damping

constant a = (6VL/AVD) Jin 2 (where aVL is the Lorentz half width in

sec.-1) have been tabulated by Poesner (33). The value of a for Na

5890 A. was obtained from Hinnov and Kohn (22). Although the value of

a for Cd 2288 A. is not readily available, its value is estimated to be

about the same as that given for Ag 3281 A. by Hinnov and Kohn (22).

Other listings of a values are to be found in James and Sugden (23),









Sobolev (37) and Mitchell and Zemansky (31). A fuller discussion of

the shape of spectral lines and its effect on atomic absorption is con-

tained in Appendix C. The calculation of p is discussed in Appendix D.

The results given in Table 4 in atoms/cm.3 can be converted to

solution concentration by the use of equation [21]. A minimum detect-

able concentration of 5.6 x 109 atoms/cm.3 of flame gases (for Na 5890 A.

at 28500K. (43) and the conditions given in Table 4) corresponds to a

solution concentration of about 0.5 p.p.m. The terms in equation [21]

have been taken as follows: Q = 125 cm.3/sec., nT/n298 = 1.2,

0 = 2 cm.3/min., 6 = 0.5, and = 0.04. If the flame cell length

were increased to 10 cm. the limit of detectability would drop to 0.05

p.p.m. and if the cell length were 100 cm., the limit would be 0.005

p.p.m. (13). Flames in which compound formation and ionization are less,

will result in a value of P greater than 0.04 and, therefore, a lower

limit of detectability. The limit of detectability calculated for a

10 cm. long flame compares quite favorably with measured values in the

literature (3,14) for slightly different flame conditions.

In Table 5 the shape of the Nm versus slit width, W, curve is

illustrated for several sets of conditions. Because the shape of the

Nm versus W curve is independent of the spectral line, it is not necessary

to evaluate all factors in equation [42]. The terms outside the square

root have been combined in a constant term Ks, and equation [42] has been

rewritten in the form

( 2BMecLf f
Nm = Ks TfWH(A/F2)Io + 2f + (cRd/io)2W2) 1/2 [46]


where s has been replaced by RdW from equation [2 ].











VARIATION


Monochromator
Flame
Type

W (cm.)

0.02

0.2

2.0

20

200


TABLE 5

OF Nm WITH W FOR SEVERAL MONOCHROMATORS AND FOR
SEVERAL FLAME TYPESa


Jarrell-Ashb DUC Dud


H2/02e

10-2

10-2

10-2

10-2

10-2


C2H2/02f

10-2

10-2
10-2


10-2

1.2 x 10-2


H2/02

10-2

10-2

10-2

10-2

10-2


C2H2/02

10-2

10-2

10-2

1.1 x 10-2

4.5 x 10-2


H2/02g

10-2

10-2

10-2

10-2

1.1 x 10-2


aB = 1.5; M = 106; e 1.6 x 10-19 could ; 1 = 8.9 x 10-4 watts/cm.2
ster.; Y= 5.3 x 10 amps./watt at 400 mp, 1.2 x 104 amps./watt at
600 mp; Tf = 0.5; H = 1 cm.; A = 25 cm.2; F = 50 cm.; X= 10-2; =
0.005; s = RdW.

bRd = 16 mp/cm.

CRd 100 mp/cm. at 400 mp.

dRd = 350 mp/cm. at 600 mp.

ec = 1.3 x 10-8 watts/cm.2 ster. mp at 400 mp.

flc = 4.0 x 10-7 watts/cm.2 ster. mp at 400 mp.

gIc = 1.0 x 10-8 watts/cm.2 ster. mp at 600 mp.

hic = 6.8 x 10-8 watts/cm.2 ster. mp at 600 mp.

(All Nm values in the above table have been divided by the constant

factor Ks.)


C2H2/02h

10-2

10-2

10-2

10-2

2.8 x 10-2


b









In these calculations typical values of B, M, and Y are taken for

a 1P28 photomultiplier, and 10 is estimated to be approximately the

value given by Crosswhite (11) for a typical line of an iron HCDT.

The values of X and f have been estimated as shown in the table, and

Ic has been taken from the work by Gilbert (17). For the chosen con-

ditions, W would have to be unreasonably small before the first term

in equation [46] would become significant. For this reason Nm shows

no increase as the slit width is decreased below the calculated optimum

value (Wo = 0.19 cm.). However,' slit widths below 0.19 cm. will result

in a smaller photodetector signal and correspondingly greater chance for

electronic noise interference. As the slit width is increased beyond

the optimum value, Nm increases very slowly for the conditions chosen,

due to the increase in the last term in equation [46], and very wide

slits could be used before the increase would become significant. In

this case the slit width used would be determined by the mechanical

slit width available on the instrument or by the proximity of other

spectral lines. In such instances interference filters can be used with

excellent results.

The difference in results between the Jarrell-Ash and the Beckman

DU is due only to the difference in reciprocal linear dispersion in the

two instruments. The variation in results with flame type is due only

to the variation in flame background intensity. The variation with

wavelength is a result of two competing factors. The flame background

intensity is lower at 600 mi than at 400 mp by about 6-fold for an

oxyacetylene flame, but the reciprocal linear dispersion of the DU is

about 3.5 times higher at 600 mp than at 400 mp. The result is a

slower increase in Nm at 600 mp than at 400 mp for the oxyacetylene





41


flame. For the oxyhydrogen flame the results are reversed since the

flame background intensity is about the same at both wavelengths, and

the higher reciprocal linear dispersion leads to a faster increase of

Nm with W at 600 my than at 400 mp.

The effect on Nm of varying other factors than those discussed

above can be determined by examination of equation [42].

In Appendices C, D, and E some of the details of the preceding

derivations and assumptions are discussed to a greater extent.














IV. CALCULATION OF OPTIMUM CONDITIONS

Introduction


A large number of factors affect the signal and the noise in

both atomic emission and atomic absorption flame spectrometry, as may

be seen from the signal and noise expressions presented in Sections II

and III. At concentrations other than the minimum detectable, the

expressions of interest for atomic emission flame spectrometry are:


i = jTfWH(A/F2)nI [47]


and


AiT =


[2BMecAf(id + YTfWH(A/F2)Ics) + (yTfWH(A/F2)AIc5s)2,f]/2 [48]


The expressions of interest for atomic absorption flame spectrometry

are:


io it = rTfWH(A/F2)Io( ek L) [49]


and


---T [50]
2 2BMecI --- 2 1/2
[yrTfWH(A/F2) 2f y2BTecIW + A 2 + (Ics)2 + (Is)2


However, from a practical point of view only a few factors are

important in the discussion of optimum experimental conditions. The









analyst usually begins with a specified monochromator-detector-readout

combination, which places such factors as Tf, A, F, and Af beyond his

control. The variables which the analyst is most often interested in

optimizing are monochromator slit width, W, flame type (e.g., C2H2/02

or H2/air), and flame conditions (gas flow rates and solution flow

rate).

If it is assumed, as in the preceding sections, that the entrance

and exit slit widths are equal, then the relationship of the slit width

to the signal and noise is evident from the above equations. The other

factors affect the signal and the noise in a number of ways which are

not immediately apparent. It may be seen that the flame background

intensity Ic (or AIc), which is dependent on flame type, affects the

noise, AiT, in both atomic emission and atomic absorption flame spec-

trometry. Qualitatively it may be noted that the signal in both atomic

emission and atomic absorption flame spectrometry is a function of the

atomic concentration of the species of interest, N, in atoms/cm.3 of

flame gases, and the atomic concentration is dependent on compound

dissociation and ionization, which depends on flame composition and

flame temperature. In this section expressions will be derived relating

the signal and the noise to flame type and flame conditions.



Selection of Optimum Slit Width


An expression for the optimum slit width as a function of experi-

mental parameters for atomic emission and atomic absorption flame

spectrometry may be obtained by maximizing the respective signal-to-

noise ratios, i.e., differentiating the signal-to-noise ratio with








respect to W, equating to zero, and solving for the optimum slit width,

Wo. The signal-to-noise ratio for atomic emission flame spectrometry

is obtained by dividing equation [47] (l= 1) by equation [48].


i/AlT =

YTfWH(A/F2)nI
[51]
[2BMecAf(id + YTfWH(A/F2)Ics) + (tTfWH(A/F2)A cs)2f] 1/


The optimum slit width for atomic emission flame spectrometry is most

readily obtained by finding the minimum of the noise-to-signal ratio by

differentiating the square of the inverse of the above equation, i.e.,


(<^/i)2 = (klW)-2 [k2(id + k3Ws)


+ (k4Ws) 2f] [52]


where kl = fTfH(A/F2)nI ,

k2 = 2BMecAf ,

k3 = 'TfH(A/F2)Ic ,

k4 = ITfH(A/F2)Ac.

If s is given with sufficient accuracy by RdW, where Rd is the reciprocal

linear dispersion of the monochromator in mp/cm., then the optimum slit

width is found to be given by


[ 2BMecid I /4 [53]
0 = (XTfH(A/F2)c(f) /2Rd) [53]


This is the same expression as previously dervied for the opti-

mum slit width at the limit of detectability expression with respect to

W. The case in which s is not given with sufficient accuracy by RdW

has also been discussed in Section II.








The signal-to-noise ratio for atomic absorption flame spectrometry

is obtained by dividing equation [49) by equation [50].

(io it)/AIT [54]

YTfWH(A/F2)IO(1 e-PkoL)
2)j2 2eo 2 2 2 1/2
[fyTfWH(A/F2) BfeTWHA/F2 + Go)2 + Acs) + (Als)2]

As in the above case for atomic emission, the optimum slit width

expression is most readily obtained by minimizing the noise-to-signal

ratio by differentiating the square of the inverse of equation [54], i.e.,


/(io ic)] 2 = [55]


(k5W)-2 [ (k6W)2Af [k7W1 + (o)2 + (cs) + (I)2 ]

where k5 = 'TfH(A/F2)IO(1 e-kOL)

k6 = lTfH(A/F2) ,

k7 = 2BMecIO/(YTfHA/F2).

The exact expression for s may be substituted into equation [55] without

overly complicating the results, i.e., s = (Rd22 + sc2) /2. The

various expressions for spectral slit width have been more fully dis-

cussed in Section II. If the indicated operations are carried out after

substituting for s in equation [55], the optimum slit width expression

is found to be


BMec0o 1/3
o = [ cRd) 2TfHA/F2 [56]


As might be expected this is the same expression as previously

obtained by minimizing the limit of detection expression for atomic

absorption flame spectrometry with respect to slit width.









Effect of Flame Conditions on Atomic Concentration


The selection of optimum flame conditions is a somewhat more

difficult problem than the selection of an optimum slit width. The

origin of this difficulty resides in the nature of the flame source.

The temperature of the flame source and the composition of the flame

gases affect the atomic concentration of the species of interest through

compound formation and ionization, but the flame temperature and flame

gas composition cannot be chosen independently. There exists for any

given flame a relatively narrow range of conditions of gas flow rates

and fuel-to-oxygen ratios over which it is a stable, analytically useful

flame, and for any given flame there is a relatively narrow temperature

range (43) over which it can be adjusted. Therefore, an optimum tem-

perature cannot be chosen in the same way that an optimum slit width

can, because a major change in flame temperature involves a change in

flame type (e.g., from C2H2/02 to H2/02), with corresponding changes in

Ic, AIc, and compound formation of the species of interest with flame

gas constituents.

To find the optimum flame conditions the procedure which will be

followed is to choose a particular flame and calculate the variation of

the atomic concentration of the species of interest, N, and the variation

of the signal-to-noise ratio over the temperature range which that flame

can exhibit. The calculation of N and the signal-to-noise ratio then

will be repeated for other flames. For both atomic emission and atomic

absorption flame spectrometry, plots of signal-to-noise ratio versus

temperature for several flame types will allow the selection of the

optimum flame type for the particular analysis of interest.









The atomic concentration in the flame of the species of interest

is, of course, related to the concentration of the solution aspirated

into the flame, as well as being related to compound formation and

ionization. No derivation of N is meaningful unless N can also be

related to the solution concentration C. An expression has been derived

in Section II (equation [21]) relating N and C. In slightly altered form

this equation reads


N/P = 3 x 1021 n298C [57]
nTQT


In the following derivation it will be more convenient to relate C to

PT, the total pressure in atmospheres of the species of interest present

in all forms, atomic, molecular, and ionic. The term N/P is the total

concentration, NT, in particles/cm.3, of the species of interest in all

forms. Because NT is small compared to other species for most flame

spectrometric studies, NT may be related to PT by means of the ideal gas

expression (30), PT = NTkT, where PT is in atmospheres, k is the

Boltzmann constant in cm.3-atm./OK. (1.38 x 10-22), and T is the flame

temperature in OK. The desired expression is, therefore,



P, 3 x 10k n298GC [58]
T QnT



For a given value of PT, the equilibrium concentration of atoms of

the species of interest present in the flame is dependent upon the

dissociation of the aspirated salt, the formation of compounds with the

flame gases, and ionization. If equilibrium is assumed, N can be pre-

dicted by consideration of the above processes. For the case in which

an aqueous solution of sodium chloride is aspirated into the flame, the









formation of only one flame gas compound, NaOH need be considered

(24).

Sodium chloride has been chosen as the aspirated salt because

this is probably the most common means of introducing sodium into a

flame, although no change in the results of the theory occur if NaN03,

NaC104, or other similar salts are considered. Most sodium salts,

except NaOH, are completely dissociated at the temperature of flames

used in flame spectrometry. The following equilibria expressions and

their corresponding material balance expressions may be written for

sodium aspirated as sodium chloride.

The dissociation of the aspirated salt,


NaCl -- Na + Cl ,


has an equilibrium constant K1 given by



K1 = PNaPC1 [59]
PNaC1


where PNa' PC1' and pNaCl are the partial pressures in atmospheres of

Na, Cl, and NaCI, respectively. Assuming that Cl in the flame comes

only from the dissociation of the salt, and that the pressure of HC1

formed in the flame gases is a small fraction of the total pressure of

Cl in the flame, and further assuming that PNaC1 < PT, then pCI is

given by


PC1= PT PNaC1 T [60]

When using H2/02, H2/air, and C2H2/02 flames, the above assumptions









are valid. The stability of HCI is low (25), and so dissociation (see

Appendix F) should be nearly complete. The stability of NaC1, as will

be evident, is also low, and so the assumption PT >> PNaCl is also

valid. Substituting for pCl in equation [59] and solving for PNaC1

gives


PNaC = PNaT [61]
Kl


For the ionization reaction


Na -- Na+ + e ,


the equilibrium constant, K2, is given by


K2 = PNa+Pe [62]
PNa


where PNa+ is the partial pressure in atmospheres of sodium ions in

theflame, and Pe is the partial pressure of electrons in the flame.

The partial pressure of electrons in the flame is the sum of the

partial pressures of electrons due to ionization of the flame gases

and electrons due to ionization of the metal, i.e.,


e = Pe + PNa+ [63


where Pe is the partial pressure in atmospheres of the flame gas

electrons. Substituting for Pe in equation [62] gives



K2 = PNa+Pe + PNa +264
2 PN [64
PNa








Solving for PNa+ by the quadratic expression gives


-Pe + (Pe2 +
PNa =


4K2PNa)1/2


[65]


For the dissociation reaction of the flame gas compound NaOH,


NaOH ---


Na + OH ,


the equilibrium constant K3 is given by


K3 = PNaPOH
PNaOH


where pOH and PNaOH are the partial pressures

and NaOH, respectively, present in the flame.


PNaOH = PNaPOH
K3


Because PT is the total pressure of the

sent in all forms, PT is given by


in atmospheres of OH

Solving for PNaOH gives


[67]


species of interest pre-


PT = PNa + PNa+ + PNaC1 + PNaOH


[68]


It may appear that the forms NaH and NaO should also be present in

equation [68], but it was theoretically verified from dissociation

constants of NaH and NaO that even in flames of relatively large PH

and pO, the NaH and NaO species are essentially completely dissociated.

The influence of H atoms from the flame gases on the formation of HCI

in the flame has not been considered because, as will be seen, the


[66]









dissociation of NaCl is essentially complete, and this effect will not

produce any changes in the results of the theory. In Appendix F the

effect of introduction of an excess of Cl into the flame is considered

i.e., introduction of NaC1 into the flame in the presence of HC1. Sub-

stituting for PNaC1' PNa+' and PNaOH from equations [61], [65], and

[67], one obtains

2 1/2
PT PNa + -Pe + (Pe + 4K2PNa) PNaPT + PNaPOH [69]
2 KI K3


Clearing fractions and combining terms, one obtains


2(KlK3 + K3PT + KlPOH)PNa + K1K3(Pe2 + 4K2PNa)1/2

KK3(2PT + Pe) = 0 [70]


In order to use equation [70], the equilibrium constants K1,

K2, and K3 must be known at several temperatures over the temperature

range of the flames of interest. The partial pressure of OH, POH, and

the partial pressure of flame gas electrons, p must also be known for

the flames of interest. These factors, pOH and pe, are characteristic

of the particular flames and vary somewhat with temperature. However,

in the calculations to follow, pOH and Pe will be assumed to be constant

over the temperature range of a particular flame.

Solution of equation [70] for pNa for a particular flame and a

number of values of PT is somewhat tedious (a graphical method seems to

be the easiest), but fortunately two limiting cases which are of practical

importance may be noted in which equation [701 can be greatly simplified.

In most of the calculations reported in this paper, one or the other of

the simplified forms can be used.








In the first case, if ionization of the sodium atoms is unimpor-

tant, i.e., pe2 >> 4K2PNa, then equation [70] becomes


(K1K3 + K3PT + K1POH)PNa KK3PT = 0 [1]


and equation [71] is readily solved for pNa


KIK3PT
PNa = [72]
PNa K3 + K3PT + KlPOH [72]


The second case occurs when the partial pressure of flame gas

electrons becomes negligible when compared to those produced by ioniza-

tion of the metal. In this case (4K2PNa)/2>> Pe and equation [70]

becomes

1/2 1/2
(K1K3 + 2K3PT + 2KlPOH)PNa + KlK3K2 PNa

2KKPT = 0 [73]


Equation [73] is in the form of a quadratic equation in which Na/2

is the variable. Solving by the quadratic expression and squaring

gives


PNa = [74]

F-KK3K2 1/2 + [(KlK3K21/2)2 + 4(K1K3 + K3PT + KIPOH)KlK3PT] 1/2 2
2(KlK3 + K3PT + KIPOH)

The partial pressure in atmospheres of Na, PNa, can be converted

by the ideal gas expression (30) to N, the atomic concentration of Na
3
in the flame in atoms/cm. i.e.,


N = PNa/kT ,


[75]









where k is the Boltzmann constant in cm.3-atm./K. Therefore, multi-

plying either equation [70], equation [72], or equation [74] by 1/kT

allows the calculation of N as a function of temperature for a number

of flame types and a number of values of PT (and because PT is related

to the solution concentration by equation [51], N can be calculated at

any value of solution concentration).

The results of the calculation of N are shown in Figure 1, where

log N is plotted versus T for seven values of PT and three different

flame types. From Figure 1, the effects of compound dissociation and

ionization may be seen, as well as how these competing factors change

with PT and with flame type. As PT increases the maxima of the curves,

i.e., the temperatures at which ionization becomes more important than

further compound dissociation, shift to higher temperatures. As p0H

decreases, e.g., in going from the C2H2/02 flame to the H2/02 flame,

the maxima of the curves are seen to appear at lower temperatures.

In all the calculations whose results are shown in Figure 1, the

compound NaC1 was found to be completely negligible with respect to NaOH.

The results shown in Figure 1 would be exactly the same if other salts,

such as NaNO3, and NaC104, with dissociation constants similar to NaC1,

were used for the aspirated species.



Optimum Flame Conditions for Atomic Emission Flame Spectrometry


Figure 1 is very useful for the discussion of the effects of com-

pound formation and ionization, but the signal-to-noise ratio is needed

for the selection of optimum conditions. The signal for atomic emission

flame spectrometry is given by equation [47]. In seeking the optimum


































o
X















II



































o
Pf



























E-4

.4
,-4

.4


*I
Hr


^0
I

I-
x
I
.
\

0
i-4
I


x
O I *
1 <


01

/-
/
x
.4
01


























0
i-4

x

.4
*4


-I


/
/ x

/ ^


4-J




n)
oo





II
II










c a
0 o













\4







O o

o
oo











0 0o
0

a &



















O C)






II


a2. 0.


SI I I I
O
S-4 <0-- --I
i- -


--I

o
4-1
0

E-4
s-I





s-I

















el



0
(a





iCl

-J




cr




































m
U)
U)
H







-4







0
































a o
En






0
r.
.re
t0









,--



0U
U


r-i


oc'



CO
'4-

0


r4





,4 U

-4U2

C 04
CO
4J 5-1








flame conditions the instrumental parameters, except W, will remain

constant because the object sought is the flame type which will give

the largest signal-to-noise ratio for a given instrument. Therefore

equation [47] may be rewritten, using the optimum slit width expression,

equation [53], to give


i = kg 1c)-1/2I [76]


where k TfH(A/F2)n [2BMecid 1/4
8 T(ITfH(A/F2) (af)1/2Rd)2J

In the evaluation of k8 it is assumed that 1 is unity.

The relative signal for each flame type can then be calculated by

substitution of the proper value of lc and the proper integrated in-

tensity expression into equation [761. However, the evaluation of I,

the integrated intensity, is complicated by the fact that a plot of I

versus sodium concentration would show two distinct regions of the curve.

(See Appendix G for a further discussion of the intensity of spectral

lines.) In the low concentration region the integrated intensity is

proportional to the first power of N and is in fact given by


hVogu
I = o4 NAtLe-Eu/kT [77]
1074B (T)


where all the terms have been previously defined. In the high con-

centration (or self-absorption) region, the integrated intensity in

watts/cm.2 ster. is proportional to the square root of N and is given

by

hvo2 -E /kT( LQDgu 1/2
I = 7 e u n aNAtL [78
o 107 -I-In 2 B(T) '







There is, of course, a small intermediate region in which the

integrated intensity expression corresponds to neither of the above

expressions. However, N can be converted to integrated intensity

values with sufficient accuracy by extrapolating from the extremes to

find the point of intersection of the two regions (i.e., some value of

N which would give the same value of I by either expression), and using

equation [77] for N values below this point and equation [78] for N

values above this point. If equations [77] and [78] are equated at

N equals Ni, then the intersection point, Ni, is found to be given by


16tAVD B(T)
Ni n 2 AtLXogu [79]


Therefore, for N values less than Ni the detector output signal for

atomic emission flame spectrometry, i, can be calculated by the

expression


i = k8k9Ic)-l/2LNeu/kT, [80]


i.e., equation [77] is introduced into equation [76], and

hQogu
k9 = 1074tB(T) At

For N values greater than N. the detector output signal can be calculated

by the expression


i = kgklO(Ic)-1/2L1/2N1/2e-EukT [81]


i.e., equation [78] is introduced into equation [76], and

h90o2 ( ADgu ) 1/2
k10 = 107c n B(T) aA)
cc c 'ln B (T)








For convenience in further calculations, the constant kl0 can be

written in terms of the constant k9. Of course, continuity exists at

N = Ni, and so equation [80] and equation [81] will give the same value

of i at the atomic concentration Ni, i.e.,

k8k9Ic)-/2LNieEu/kT= -k8kl0 c)-1/2L1/2Nil/m2eEu/kT [82]


and solving for kl0 gives

1/2 1/2
kl0 = k9L /Nil [83

Substituting from equation [83] for kl0, equation [81] can be re-

written to give

S -1/2 1/2 1/2 -Eu/kT
i = k8kg(-Ic) LNil 2- [84]


Equations [80] and [84] are altered slightly if more than one

spectral line of the species of interest lies within the spectral slit

width of the monochromator and, therefore, reaches the detector. This

situation arises in the case of the sodium resonance doublet (5890,

5896 A.), which is not resolved by the Beckman DU monochromator. The

intensity of each of the lines making up the double is still given by

equation [77] for the low concentration case and by equation [78] for

the high concentration case.

If 1I is the intensity of line 1 (5890 A.), and gl is the

statistical weight of the upper state of line 1, then for low con-

centrations

hQ0g1 -Eu/kT [85
I1 =1074iB(T) NAtLe-








If 12 is the intensity of line 2 (5896 A.), and g2 is the statistical

weight of the upper state of line 2, then for low concentrations


hO082 -Eu/kT
12 hVg2 NALeu/kT [86]
1074tB (T)


Because Vo, At, and E are essentially the same for both lines, the same

values can be used in both equations [851 and [86] without introducing

significant error. For the high concentration case the intensities of

lines 1 and 2 are given by


hVo 2 -Eu/kT 1Dgl 11/2
hI 2 aN)t [87]
S107c Eu/T ln B(T) t

hVo2 eEu/kT AD2 1 /2
12 = n07c ( n B(T) aN [88]


Equation [76] for the detector output signal for the case of an

unresolved doublet is then given by

i = k8GIc) -1/2 ( I1 22) [89]


where V1 is the slit function factor for line 1 and K2 is the slit

function factor for line 2. The evaluation of k for each spectral

component can be carried out by use of equation [6] in Section II, i.e.,


= 1 i - ol [6]
s

For the Na 5890, 5896 A. doublet, when using the Beckman DU mono-

chromator (Rd = 330 mp/cm. at 5890 A.), if the wavelength setting

of the monochromator is 5892 A. (nearer the stronger line), and if the








monochromator slit width is 0.01 cm., then s = RdW = 3.3 mp, Y1

0.94, and 2 = 0.88.
Substituting the proper intensity expressions into equation [89]
and collecting terms shows that for low concentrations the detector
output signal for atomic emission flame spectrometry is given by

h^ A
i = k8 ) -1/2 hOAtT NLe-Eu/kT( g + 2g2) [90]
i= 80,1c) 1074cB (T) eigl1+1 K292) 90]


Collecting constant terms gives

i = kgkll( c)-1/2NLe-Eu/kT, [91].

where
hO A
ot
kll = 10743B(T) (lgl + 2g2)


At high concentrations the detector output signal is given by

h0o2 eEuT D aNAtL 1 /2 12
i = k__^1/E12 ) 1


+ k2 (g2) ] 1 [92]


Collecting constant terms gives

i = k8kl2 c -1/2 1/2 1/2 -E/kT [ 93]


where

h 2 D A A \1/2 /2
o t [ (g 1/
kl12 107c \ n 2 B(T) [ 1 2l +2








By equating equation [91] to equation [93] when N equals Ni, it is

possible to evaluate k12 in terms of kll, i.e.,


k2 = k lNil2L1/2 [94]


and i is then given by


i = k8kll -c)-1/2Nil/2N1/2Le-EukT [95]


Equation [79] for the value of N at the intersection point, Ni,

is also altered for the case in which the doublet is not resolved. In

this case'Ni is given by


16B (T)AVD a[l (gl) 1/2 + 2(g2) 1/2 2
N =[96
i AtLXo2 V1-2 (~8gl + Vg2)2


It should be noted at this point that the optimum slit width for

atomic emission flame spectrometry was derived with the assumption that

the spectral line of interest was single, sharp, and isolated. This

assumption is not valid for the analysis of Na using the Beckman DU

monochromator. The value of I in equation [51] should be replaced by

I = ii + V212 Because 1i and 12 are dependent on the slit width,

W, I is in this case dependent on the slit width. Fortunately the

dependence of I on W is slight because 11 and L2 are close to unity

(see example of calculation of i and (2 above), and the dependence

of I on W will always be slight when the spectral lines are close

together, and the spectral slit width, s, is large.

Calculation of the total root-mean-square photodetector output

noise, AiT, for atomic emission flame spectrometry can readily be

accomplished by assuming that Ic ( and the fluctuation in Ic, AIc) is









approximately constant over the temperature range of any given flame.

This assumption should not affect the results of the prediction of the

optimum flame type, because the variation of I over the temperature

range of a given flame is quite small, whereas the variation of I
c
in going from one type flame to another is quite great (17).

When the proper optimum slit width is used in equation [48] for

calculating AiT, the value of AiT will be approximately the same for

each flame type. Qualitatively this could have been anticipated because

optimization of the slit width results in obtaining the slit width at

which the noise due to flame flicker (YTfWH(A/F2)AIc(Af)1/2s) is reduced

to approximately the same value as the noise due to the dark current
1/2
shot effect ([2BMecid] 1). A decrease in the slit width below the

optimum value does not result in an appreciable decrease in the noise,

because the dark current noise does not depend on W. Because for a given

detector the dark current is the same no matter which flame is used,

AiT is approximately the same for all flame types if the optimum slit

width is used. Table 6 shows values of W1 and AiT for three flames,

calculated using the values listed under the table, which are typical

values for a Beckman DU monochromator at 5890 A., a 1P28 photomultiplier,

and a total consumption atomizer-burner.

Because AiT is approximately the same for all three flame types,

the flame type giving the maximum signal will also give the maximum

signal-to-noise ratio. For low concentrations where the signal, i, is

given by equation [80], the signal-to-noise ratio is given by


i/Ai = kclIc)-/2LNeEuT [97]



















TABLE 6

OPTIMUM SLIT WIDTH AND TOTAL ROOT-MEAN-SQUARE NOISE
FOR THREE FLAMES


Flame Wo (cm.) AiT (amps.)

H2/air 6.1 x 10-2 3.2 x 10-10

H2/02 1.7 x 10-2 3.1 x 10-10

C2H2/02 6.5 x 10-3 3.2 x 10-10


The following values were taken as typical for a Beckman DU monochromator
and a 1P28 photomultipl er: B = 1.5, M = 106, ec = 1.6 x 10-19 coulombs,
i = 10-7 amps., = 10' amps./watt, Tf = 0.5, H = 1 cm., A = 25 cm.2
F = 2500 cm.2, Rd = 330 mp/cm., Af = 1, AlI = 0.005 I, I = 7.7 x 10-10
watts/cm.2 ster. mp for H2/air (17), Ic = 9.7 x 10 watts cm.2 ster. mp
for H2/02 (17), and Ic = 6.8 x 10-8 watts/cm.2 ster. mp for C2H2/02 (17).








where kc = k8k9/LiT. For high concentrations where the signal, i, is

given by equation [841, the signal-to-noise ratio is given by


i/iT = kc c)-1/2LNil/2Nl/2e-Eu/kT 9[8]


For the case of the unresolved doublet spectral line (equations [91]

and [95]), at low concentrations the signal-to-noise ratio is given by


i/AiT = kd(--c)1/2NLe-Eu/k [99

where kd = k8kll/iT, and at high concentrations the signal-to-noise

ratio is given by


i/ai = kd ) 1/2i/2N1/2Le-EukT [100]


Figure 2 shows plots of i/aiTkc versus T. The upper dashed line

in Figure 2 is i/ZiTkc at N = Ni, where Ni has been calculated from

equation [79]. Below this dashed line k/AiTkc is calculated from

equation [97]. Above this dashed line i/i-Tkc is calculated from

equation [98. At low concentrations i/AiTkc = i/i-Tkd

(Lc)- /2LNeEu/kT, i.e., the curves given in Figure 2 are valid whether

a single spectral line or a multiple passes through the spectral slit

width. In this low concentration region the plotted signal-to-noise

ratios are entirely independent of the instrument used, i.e., the

optimum flame type can be determined without knowing what instrumental

setup is to be used. The absolute value of the signal-to-noise ratio for

a particular instrumental setup, flame type, and flame temperature can

be calculated by evaluating kc (or kd) by substituting values for the

experimental parameters.

















a'


0


-I
0
'-
xl

-zj


I
0
1-4


-4
0
-4
xl


CO






'I
C-


SI I I r


0
0
c-1


u






0 <'-
-4l


0
J0


I-4









-o

















*4
3








00
1-I
oo





















-1
o0













Zr-e
0
L

















U0
inm
























t
0



" EW






,-4 1
o .U




-40
4 C/























0


C'I 4-1
cfE










At high concentrations the plotted signal-to-noise ratio depends

to a small extent on the instrumental setup because i/AiTkc depends on

Ni, the value of which depends on whether the doublet is resolved or

unresolved. When the spectral line is single (as the case would be

when using the 0.5 meter Ebert mounting, Jarrell-Ash monochromator), Ni

is calculated from equation 179]. When the spectral line is an un-

resolved doublet, Ni is calculated from equation [96]. The difference

in values of Ni as calculated from equations [79] and [96] is small

enough that neglecting the instrumental dependence of Ni will not affect

the choice of the optimum flame type. This is especially true because

the signal-to-noise ratios i/AiTkc or i/AiTkd depend only on the one-

half power of Ni.

Neglecting the small instrumental dependence of the signal-to-

noise ratios plotted in Figure 2, much information about optimum con-

ditions for analysis of Na by atomic emission flame spectrometry may be

obtained from Figure 2. The two dashed lines for each flame mark the

upper and lower limits of an optimum region for that flame. The upper

dashed line (S.A.), as previously discussed, indicates the beginning of

the self-absorption region. Above this line the signal, and hence the

signal-to-noise ratio, is proportional to the square-root of N, the

atomic concentration. The lower dashed line (L. of D.) marks the limit

of detectability for the Na 5890, 5896 A. doublet for the particular

flame when using a Beckman DU monochromator and a good 1P28 photo-

multiplier. The limit of detectability lines were obtained by calcu-

lating the limits of detectability in atoms/cm.3 for Na in each of the

three flames, at several temperatures, from equation [16]. From these








values of N, i/i-Tkc values were calculated using equation [97]. The

limit of detectability lines are the only lines on Figure 2 which depend

to a great extent on the instrumental setup. The limit of detectability

will be different with each monochromator-detector combination, and

therefore this line must be calculated for specific cases.

In the region between the self-absorption line (S.A.) and the

limit of detectability line (L. of D.), the spectral line is detectable

and self-absorption of radiation is negligible. In the H2/02 flame at

25000K. the point at which self-absorption becomes important corresponds

to a PT value of approximately 4.4 x 10-6 atmospheres. At an aqueous

solution flow rate of approximately 2 cm.3/min. and gas flow rates of

2860 cm.3/min. 02 and 10,000 cm.3/min. H2, this corresponds to a solution

concentration of approximately 2.3 x 10-3 M, i.e., 53 p.p.m. (n298/nT =

1.2, = 0.5). Also, in the H2/02 flame at a temperature of 25000K.,

linear analytical curves for Na, free from the effects of self-

absorption and ionization should be obtained for PT values from

approximately 10-11 to 10-6 atmospheres (or solution concentrations of

approximately 5.2 x 10-9 to 5.2 x 10-4 moles/liter). A H2/02 flame at

a temperature of 25000K. should also give linear analytical curves for

Na but with half the slope of the previous curves due to self-absorption

for PT values above approximately 4 x 10-6 atmospheres.



Optimum Flame Conditions for Atomic Absorption Flame Spectrometry


The optimum conditions for analysis by atomic absorption flame

spectrometry can be determined in much the same way as for atomic

emission flame spectrometry. The signal-to-noise ratio for atomic








absorption has'been given in equation [54]. After dividing both

numerator and denominator by rTfWH(A/F2)I, equation [54] can be

rewritten as


(io it)/ZT =

1 e-pkOL
S T2BMec 1i /2
Af o + (0fo/Io)2 + (Zgcs/IO)2 + (sL/Io)2 [1101]
arWH(A/F2s [loll

When the optimum slit width, Wo, is used in equation [101],

(-I-cs/Io)2 is much smaller than (7o/I)2 and can be neglected. Sub-

stitution of values in the first term of the sum in the denominator

of equation [101], shows that for any practical set of conditions this

term will always be quite small when compared to (A70/Io)2, which will

never have a value smaller than about 10 4. The last term, (Is/Io)2,

will be small in most cases of practical importance in atomic absorption

flame spectrometry. The intensity of the scattered radiation Is is

always a fraction of I1. Usually the fraction is quite small, and the

scattering term can be neglected. The fraction of scattered radiation

could always be made small by using organic solvents or chamber-type

atomizers to decrease the particle size of the aspirated solution in

the flame. For the purposes of selecting the optimum flame conditions,

equation [101] can be rewritten as


(io it)/T = k3(1 e-PkoL) [102]


where


k13 = [(o/i)(f)1/2] -1







The value of ko, the atomic absorption coefficient at the line

center (in cm.-1), is given (31) by


2 2 n 2 X 02gNAt
ko = [103 ]
a0D T 8t B(T)


An expression for the Doppler half-width, AVD, has already been given

in Section III:

S2 2R In 2 T )l/2
D= --c o0 45]


After substituting AL~ from equation [45] into equation [103], and the

resulting expression for ko into equation [102], (io it)/AiT is found

to be given by

(i it)/AT = k13(l e-k4NL/T/2) [104]


where the terms which are not dependent on flame temperature or flame

composition have been collected into the constant k14, and


p2 l- 2 0o2guAticMaI/2
k4 = ,- 8rB(T) 2 v2R In 2 V 0


If the spectral line of interest is not a singlet but a doublet,

as in the case of the Na 5890, 5896 A. double, which is not resolved

by many monochromators, then equations [102], [103], and [104] must be

altered. Equation [102] should read in this case

(io it)/-T = k13(l e- k0L) [105]

where e is a factor to account for the line being multiple. The

evaluation of e is discussed in Appendix H. The value of I in the









constant k13 is the sum of the intensities of the two lines making up

the doublet, i.e., 10 = l1IOl + 2I02, where line 1 is the 5890 A.

line, and line 2 is the 5896 A. line. The value of ko is also altered.

For a doublet ko = kol + k02, where kl0 and k02 are the atomic absorption

coefficients at the line centers of line 1 and line 2, respectively.

Therefore, equation [103] should read


2 ~1n 2 x02 NAtj
ko= A~D 3-T8 B(T) (gl 8 82) [106]


where gl and g2 are the statistical weights of the upper states of lines

1 and 2, respectively, and 0o, At, and A9D are approximately the same

for both lines. Equation [104] should read in this case


(io it)/T = k13(l e-kl5NL/T/2) [107 ]


where


e 2 1n 2 02(gl g2)AtS cMal/2
k15 =
-k1 8jr B(T)2 J2R In 2


It may be seen from equations [104] and [107] that no matter which

of the two expressions most accurately gives the value of the signal-to-

noise ratio for atomic absorption flame spectrometry, when flame con-

ditions are such that NL/T1/2 has its maximum value, the value of

(io it)ZT will also be at its maximum. Figure 3 shows plots of

log (NL/T1/2) versus T for three flames and seven values of PT. These

plots have the advantage of being entirely independent of the instru-

ment used, i.e., the same set of curves can be used to determine optimum

flame conditions no matter what spectrometer is to be used in the

















oro r


II I I
0 0 0C




* x x



H-


I
o
P4
x




IaI

[-4




















x







p.'


o00
I
0

-4


'zP


I I



I
















I -
*
























I I



xI x
-4

1
Ij -j


4-)
d

E .



C, ) I
d 0
-4
C M
Ci 0
















T





-W



0 -
N U

c IOn













CO




X




E -I
SII
-



ci 0







N O- 0 "


00 ------
0
E--l



M l-1 l


O
0








C)
0

en

o


o
o
co
N





O
0







O












4-W
C)
















Sli
o o
0 0
co
N
00


Cd

O 0





r-4
o



0
0
N





0
0






0
0
0






o
-4d

















0
0
-4
O^


0




.-4
C)
co








c4




S-1



0
E-H




0
r-4
0















Cd
0
0



4d

-4



)

















E-4


z
0
0)




0







ca
a)











Cd
i-




ed









O
0
I









analysis. The absolute value of the signal-to-noise ratio for atomic

absorption can be obtained from Figure 3 by evaluating the constants in

equations [104] or [107].

The dashed lines in Figure 3 mark the limits of detectability for

the Na 5890, 5896 A. double for the particular flame when using a

Beckman DU monochromator and a 1P28 photomultiplier detector. The

limits of detectability were calculated from equation [44]. The limits

of detectability are, of course, dependent on the instrumental setup

used and must be calculated for specific cases.

Although it is not indicated in Figure 3, there is an upper limit

to the optimum region for analysis by atomic absorption flame spectrome-

try. Qualitatively it may be seen that an upper limit would be reached

when the atomic concentration, N, reached a value such that the difference

between the output signal due to the transmitted intensity when solvent

is introduced into the flame, 10, and the output signal due to the

transmitted intensity when sample is introduced, It, becomes of the same

order of magnitude as the noise. An equation for the maximum detectable

concentration can be derived in much the same manner as for the minimum

detectable concentration. However, the resulting equation is unduly

complex and can only be solved graphically. It is not particularly

important to be able to calculate the maximum detectable concentration

because it is always possible to dilute concentrated solutions in order

to work in a more favorable concentration region, and so the theory

for the maximum detectable concentration is not presented in this

paper.









Calculations


The equilibrium constant for the dissociation of NaC1, K1, as

a function of temperature, is calculated from the expression given by

Mavrodineanu and Boiteux (30),



log K = -5040 DNaCl + log T + log (1 10-0625e /T)
T 2

+ iNa iC l iNaC1 [108]


where DNaC1 is the dissociation energy of NaC1 in electron volts, WOe

is the vibrational constant of the molecule in cm.-I (21), and iNa'

iC1, and iNaCl are the chemical constants of Na, Cl, and NaC1, respec-

tively. For a monatomic gas (either Na or Cl) the chemical constant is

given (30) by


iM = -1.587 + i log Ma + log go [109]


where Ma is the atomic mass of the species of interest, and go is the

statistical weight of the ground state of the atom. For a diatomic gas

such as NaC1 the chemical constant is given (30) by


iNaC2 = -1.738 + -log NaC1 + log Mg log Be,[ll0]


where MNaCl is the molecular weight of NaC1, go* is the statistical

weight of the ground state of the molecule, and Be is the rotational

constant in cm.-1 of the molecule (21).

The equilibrium constant for the ionization reaction, K2, as a

function of temperature is also calculated from an expression given by

Mavrodineanu and Boiteux (30),









log K2 = -5040 V/T + 5/2 log T + log go' log go

6.1818, [111]


where V is the ionization energy of the atom in electron volts, go is

the statistical weight of the ground state of the neutral atom, and

go' is the statistical weight of the ground state of the ion.

The equilibrium constant for the dissociation of NaOH, K3, as a

function of temperature is obtained from thermochemical data tabulated

in the JANAF Tables (25). The values found for KI, K2, and K3 are

summarized in Table 7.

The partial pressure of flame electrons, pe, is taken to be

4 x 10-9 atm. (15) in the outer cones of all three flames. The partial

pressures of OH in the various flames are estimated from the information

given by Zaer (46), and pOH is assumed constant over the temperature

range of the individual flames. In estimating the values of POH, it is

necessary to take into account the solvent being aspirated into the

flame because the solvent contributes a large portion of the flame

gases. For the C2H2/02 flame, POH is estimated to be 0.2 atm. In the

H2/02 and H2/air flames POH is estimated to be, respectively, 0.05 atm.

and 0.001 atm.

The value of TIc is estimated to be 5 x 10-31c, where Ic has been

measured for the various flames by Gilbert (17). Atomization efficiencies

and flame temperatures are estimated from work performed in this labora-

tory (42,43). In calculating Figures 2 and 3, L is taken to be 1 cm.

in all cases, because this is found to be approximately the radius of a

flame in good adjustment. The value of Eu for Na is 2.1 electron

volts (28).

























TABLE 7

VALUES OF K1, K2, AND K3 AS A FUNCTION OF T


T (OK.) K1 (atm.) K2 (atm.) K3 (atm.)

1600 4.4 x 10-7 2.5 x 10-15 1.3 x 10-7

2000 9.1 x 10-5 6.4 x 10-12 4.1 x 10-5

2400 3.2 x 10-3 1.5 x 10-9 2.0 x 10-3

2800 4.2 x 10-2 7.4 x 10-8 3.1 x 10-2

3200 1.4 x 10-1 1.6 x 10-6 2.5 x 10-1









In calculating Ni from equations [79] and [96], the value of a

is taken from data given by Hinnov and Kohn (22). The Doppler half-

width, A D, is calculated from equation [45], and B(T) is equal to go,

the statistical weight of the ground state of Na. Spectral data, such

as At, gu, and go, are taken from the Landolt-Bornstein tables (28).



Experimental Verification of Theory


The validity of the theory developed in this paper has been shown

by comparing theoretically predicted analytical (working) curves with

experimentally determined analytical curves. The theoretical analytical

curves were obtained from Figure 2 by choosing a temperature typical of

the flame to be used (43) and then reading off PT and corresponding

lof (i/AiTkc) values at that flame temperature. Because the signal-to-

noise ratio is proportional to the signal, as previously discussed,

plots of log (i/AiTkc) versus log PT will have the same shape and slope

as the analytical curves of signal versus concentration.

The experimental curves were obtained by recording the output

signal for a wide range of solution concentrations. Solution concen-

trations were converted to Pt values by equation [581 so that measured

values of signal as a function of Pt were obtained. The measured curves

of the logarithm of the signal versus log PT were shifted along the

log (i/AiTk ) axis until the experimental curve for the C2H2/02 flame

coincided with the theoretical curve. The analytical curves for the

H2/02 and H2/air flames were shifted by the same factor as the C2H2/02

flame curve. Agreement as to shape and slope of the two sets of curves

for all three flames is quite good as shown in Figure 4.


















S0
c"!



--*o-I o *o ,,


om I II
1 0 C) o
c 00 0
I 6 0 0




f O0 U r-0' II
C 0
-Ii

cu 0 0 a'
C4 OO N
,- II 0. CO
1N II C O CM
Lj CN CN o II II o



x0 -o..



.- 0
o
5l a 0 H co





,o -



o Yo o
Q C z c LO
o E o

\ I --' * o, *. o "
S*. O C *

SO cI u II

rz co c ou c
So 0 o0 '
41 \ n r s 4

` c, c O I I h
,-O ** oC
SOrII C ) g O <
0N O II II
0 *
0 d
*-o- o

a 0


I *
0 rI

OC
S1 r0C



I \ u


SN-. o u 3
CY0 7 uH


Su0 *4
.u 0 o o-
0 cc



0 II * C-
II -- 0 CM
r0 CM *I- 0 II I
-4 I 4 :; a )H


r-









It should be noted that Alkemade (1) predicted the shape of the

sodium 5890, 5896 A. flame emission analytical curve by deriving the

"curves of growth" for the total absorption of the Na doublet. He had

good correlation between the derived curves and the experimental curves.

Alkemade (1), however, neglected the effects of ionization and compound

formation on the shapes of the analytical curves and also did not cal-

culate the intersection point of the curves. He used experimental

conditions substantially different from those used in this paper; namely,

a propane/air flame, a chamber type atomizer-burner, and a glass trans-

mission filter for isolation of the sodium doublet. Even though experi-

mental conditions were considerably different from those used in this

paper, the basic theoretical results were quite similar.

The experimental analytical curves were determined on a Beckman

DU with spectral energy recording attachment (SERA) and 1P28 photo-

multiplier detector. Efforts were made to have the measurement pro-

cedures agree as closely as possible with standard practices in order

to illustrate the practical value of the theory developed in this paper.

Analytical curves were determined for four flames with conditions as

noted on the curves. The analytical curves shown in Figure 4 are

average curves for quadruplicate measurements.

The Beckman flame housing attachment was used throughout, but the

mirror in the housing was blocked so that only radiation coming directly

from the flame entered the spectrometric system. Blocking of the mirror

was necessary to insure reproducible entrance optics. Over the wide

concentration range used in this study, the relative contributions of

radiation arriving directly from the flame and radiation reflected by









the mirror changed as the concentration changed because of the variation

of the amount of reflected radiation absorbed in passing through the

flame gases. It was found that the ratio of meter reading with mirror

unmasked to meter reading with mirror masked varied from 3.18 at 10-1

p.p.m. Na to 2.06 at 1000 p.p.m. Na. If the mirror were not blocked,

this would lead to curvature of the analytical curves not accounted for

in the calculation of the theoretical curves. However, the influence

of entrance optics could be accounted for by supplying the proper value

of n, the entrance optics factor, to adjust the measured signal values

(see equation [47]).

In determining the experimental curves the calculated optimum

slit widths were used, and the sensitivity control was adjusted to keep

the readings on scale. With the Beckman housing a large region was

observed by the monochromator (approximately 1.6 cm. wide by 2.6 cm.

high). With the setup used, the lower edge of this region was 1.1 cm.

above the tip of the burner. The temperatures indicated for the differ-

ent flames were estimated as average temperatures over the flame region

viewed (43).

The solutions used in constructing the experimental curves had
-2
concentrations of 10-2, 10 -, 1, 0, 100, and 1000 p.p.m. The solution

of 1000 p.p.m. Na was prepared by dissolving the appropriate weight of

NaC1 in distilled water. The other solutions were prepared by successive

dilutions. Lower solution concentrations were not used because of

difficulties encountered with spurious Na emission, apparently

attributable to dust particles in the air.

As may be seen in Figure 4, the position of the intersection points

and the slopes of the experimental and theoretical curves agree quite









well. Because the H2/02 flame with the lower flow rate of H2 has a

radius approximately half that of the other flames, the theoretical

intersection point for this case was calculated using L = 0.5 cm.

This flame does not represent a practical case because the size of the

flame is too small to be experimentally useful. However, it is interest-

ing to note that the theory quite adequately accounts for the effect

of flame size on the point at which self-absorption becomes important.

Note that in Figure 2, only one value of pOH is given for H2/02

flames. Measurement of the OH emission in the outer cone of the flame

at 3090 A. for both H2/02 flames shown in Figure 4 indicated that the

value of POH was approximately the same in both cases. It should not

be too surprising that pOH is the same in the outer cones of both flames,

because there must be considerable entrainment of air in the turbulent

flames produced over total consumption atomizer-burners. The OH in-

tensity measurements were performed with the flame masked by a flat

black baffle with an opening of approximately 2 mm. by 3 mm. The meter

reading for the "fuel-rich" flame was approximately twice that of the

"stoichiometric" flame, as would be expected for two flames at the

same temperature, with the same POH, and a radius ratio of 2 to 1.

Figure 4 also shows that the theory is quite adequate in predict-

ing the signal-to-noise ratios for the various flames. All the experi-

mental points were shifted along the log (i/AiTkc) axis by the same

factor, and the experimental points for each of the flames agree fairly

well with their respective theoretical curves. Therefore the measured

signal-to-noise ratios of the various flames must be related to each

other in th- same way as the theoretical signal-to-noise ratios. For









example, for log PT = -6, the theoretical values of log (i/AiTkc) for

H2/air, H2/02 (curve 1), and C2H2/02 are 12.3, 12.5, and 12.0, re-

spectively, while the experimental values are 12.1, 12.6, and 11.9,

respectively, i.e., the theoretical values of the three flames are in

the ratio of 1.02/1.04/1.00, while the experimental values are in the

ratio of 1.02/1.06/1.00, approximately the same, within experimental

error.

The theoretical curve for the "stoichiometric" H2/02 flame (curve

2) has been adjusted to account for the size difference. Because the

region viewed by the spectrometric system is wider than the flame width,

the signal should increase as the square of the flame radius in going

from the "stoichiometric" to the "fuel-rich" flame. The "fuel-rich"

H2/02 flame has a radius of 1 cm., the "stoichiometric" H2/02 flame has

a radius of 0.5 cm., and, therefore, the size correction should be a

factor of 4. There is also a small temperature difference between the

two H2/02 flames. In the smaller flame the temperature must be averaged

over almost the entire flame, while in the larger flame the temperature

must be averaged over a smaller portion of the flame. (The smaller

flame was approximately 4 cm. long, while the larger flame was approxi-

mately 8 cm. long.)

The excellent agreement of the theoretical and experimental

analytical curves also demonstrates that the influence of species such

as NaO and NaH must be negligibly small. The agreement of theory and

experiment also indicates that, contrary to many statements to be found

in the literature, the formation of the NaOH species in the flame gases

is not negligible.









The essential correctness of the equation for optimum slit width

for atomic emission flame spectrometry, equation [53], has been verified

by measuring the noise as a function of slit width. The peak-to-peak

noise was measured at several slit widths above and below the calcu-

lated optimum slit width. The peak-to-peak noise was multiplied by

1/2 42 to obtain root-mean-square noise, and the ratio of W to root-

mean-square noise was plotted versus W. For a constant intensity source,

the signal should be proportional to W (see equation [47]), and W/noise

should be proportional to signal/noise.

In Figure 5 the unbroken line indicates the calculated variation

of log (W/AiT) as a function of W for a H2/02 flame. The value of

W/AiT is given by


W/ai = W [2BMef(id + YTfWH(A/F2)Ics)


+ ( TfWH(A/F2)Afcs)2]f ] -1/2 [112]


The circles in Figure 5 indicate measured values of log (W/noise) for

a H2/02 flame. The measured values of W/noise were shifted along the

signal-to-noise axis to obtain overlap of the experimental and theoretical

values in order to compare curve shapes. Agreement of theoretical and

experimental values is adequate. The scattering of experimental points

is due to the difficulty of measuring accurately the small values of

noise. All noise measurements were made at 5800 A.

The H2/air and C2H2/02 flames gave signal-to-noise ratio versus

W plots similar to the H2/02 flame plot. A Moseley x-y recorder

(Model 135, F. L. Moseley Co., Pasadena, California) was used for all

noise measurements.








S5.6-
I 0
5.2_

a 4.8-

S4.4

S4.01 -
5 1.2 1.4 1.6 1.8 2.0

W (cm. x 102)

Figure 5. Calculated and Experimental Plots
of Signal-to-Noise Ratio Versus Slit Width
for Sodium in a Stoichiometric H2/02 Flame.



Because of the instrumental problems inherent in atomic absorption

measurements of the alkali metals and because of the greater appli-

cability of atomic emission flame spectrometry for the analysis of

sodium, no experimental measurements of absorbance as a function of

concentration (or PT) were made. However, because the theoretical atomic

emission analytical curves agreed quite well with the experimental

curves, it is highly probable that'the theoretical and experimental

atomic absorption analytical curves should also agree very well. The

emission analytical curves depend directly upon N, as do the atomic

absorption curves, and so the verification of the emission curves

indirectly verifies the absorption curves.















V. CONCLUSIONS


It is hoped that the-work presented in this dissertation adequately

demonstrates that both atomic emission and atomic absorption flame

spectrometry can be treated quantitatively in a relatively simple manner.

The equations discussed allow one to predict the effects of variation

of instrumental slit width, the effects of compound formation and

ionization in the flame, and the effects of self-absorption of radiation.

This quantitative treatment makes it possible to predict the shape of

analytical curves and, therefore, to choose conditions so as to obtain

linear analytical curves. It is also possible to calculate values of

limits of detectability, signal, noise, and signal-to-noise ratios for

any instrumental setup.

Information on optimum experimental conditions for atomic emission

and atomic absorption flame spectrometric analysis of any spectral line

of any atom of any compound introduced into any flame and measured

using any experimental setup could be presented in the form of graphs

such as those in Figures 2 and 3. With accompanying data on optimum

slit widths and limits of detectability for particular instruments, this

information should be extremely valuable to many analysts using flame

spectrometry who are not necessarily specialists in the field.

Enough data are presently available that a treatment such as has

been presented in this paper for sodium could be applied to many elements

of interest in flame spectrometry. Preliminary calculations have already









been carried out on Li, K, and Mg. The results of these calculations

will be tested experimentally and presented at a later time. The

quantitative approach presented in this paper certainly would seem to

be a faster and surer method of obtaining optimum conditions than the

trial-and-error methods presently employed.

It should be noted that the essential agreement with experiment

of the theory presented in this paper should be good indirect evidence

for the existence of chemical and thermal equilibrium in the flame,

because equilibrium was assumed in the development of the theory. The

theory of atomic emission flame spectrometry presented in this paper could

also be extended to situations in which the excitation of radiation is

non-thermal, e.g., excitation by chemiluminescence, by substitution of

the proper expression for I in equation [76]. In fact, the general

approach discussed in this paper could be extended, with good purpose,

to many other areas of spectrometry. The case of the d.c. arc source is

only one example of an area which could benefit greatly by being subjected

to the quantitative approach presented in this dissertation.






































APPENDICES













APPENDIX A

Derivation of Current-Intensity Expression


The output signal of a photodetector, i.e., anodic current, in

amperes, as a result of monochromatic radiation (spectral line half-

width is assumed to be much narrower than the spectral slit width, s)

incident on the entrance slit of the monochromator, can be derived by

consideration of entrance optics, monochromator optics, and the detector

sensitivity. If the total (integrated) intensity of a spectral line is

denoted as I and has units of watts/cm.2 of source steradian, then the

total power of radiation reaching the monochromator entrance slit as a

result of this spectral line is given by


entrance


I (area of source).(solid angle viewed by monochromator). [113]


If the monochromator entrance slit is fully and uniformly illuminated

by the radiation from the actual source, then the slit acts as the

effective source, and so the area of the source is given by the area

of the slit. The solid angle viewed by the monochromator is given by

A, the effective aperture of the monochromator, divided by F2, the

square of the focal length of the collimator, as long as the effective

aperture is fully illuminated.

Because of reflection and absorption losses within the mono-

chromator, only a fraction Tf of the radiant power reaches the exit









slit. If the exit slit is equal to or slightly larger than the entrance

slit, then the total power of radiation at the exit slit is


Pexit = Pentrance Tf watts [114]


Using the detector sensitivity factor r in amperes output at the anode

per watt of radiation incident on the photocathode, the output current,

i, from the phototube can be found, i.e.,


i = P exit amperes [115]
exit

The output current of a photodetector in amperes as a result of

polychromatic or continuous rather than monochromatic radiation in-

cident on the monochromator entrance slit can be derived in a manner

similar to the above case. However, in this particular case an addi-

tional factor must be included, namely the spectral slit width, s, of

the monochromator. This is a result of the intensity of a continuum,

Ic, being expressed as watts/cm.2 of source ster. wavelength inter-

val, and so in this case the total power of radiation reaching the

monochromator exit slit as a result of a continuum being incident

on the entrance slit is given by


Pexit = Ic (area of source)-(solid angle viewed by monochromator).

(spectral slit width) Tf watts [116]


where the spectral slit width, s, is the wavelength interval passing

through the exit slit for any particular wavelength setting. The current,

ic, due to the continuum can then be found by introducing equation [116]

into equation [115].




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs