Title: Optimization of signal-to-noise ratios in analytical spectrometry
CITATION PDF VIEWER THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00097466/00001
 Material Information
Title: Optimization of signal-to-noise ratios in analytical spectrometry the external heavy atom effect in pulsed laser time resolved phosphorimetry
Alternate Title: The external heavy atom effect in pulsed laser time resolved phosphorimetry
Physical Description: vi, 186 leaves : ill. ; 28 cm.
Language: English
Creator: Boutilier, Glenn David, 1953-
Publication Date: 1978
Copyright Date: 1978
 Subjects
Subject: Spectrum analysis -- Research   ( lcsh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 179-185.
Additional Physical Form: Also available on world Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Glenn D. Boutilier.
 Record Information
Bibliographic ID: UF00097466
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 - 000085312
oclc - 05320952
notis - AAK0661

Downloads

This item has the following downloads:

PDF ( 5 MBs ) ( PDF )


Full Text







OPTIMIZATION OF SIGNAL-TO-NOISE RATIOS IN ANALYTICAL SPECTROMETRY:
THE EXTERNAL HEAVY ATOM EFFECT IN PULSED LASER
TIME RESOLVED PHOSPHORIMETRY







By

GLE'IN D. BOUTILIER


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


1978












AC KNOW E DGEMENTS


The author wishes to acknowledge the support of the American

Chemical Society Analytical Division Summer Fellowship (1976) sponsored

by the Society for Analytical Chemists of Pittsburgh and of a Chemistry

Department Fellowship sponsored by the Procter and Gamble Company.

The author wishes to thank Art Grant, Chester Eastman, and Daley

Birch of the machine shop for construction of many of the items required

for this work. The author also gratefully acknowledges the aid of

Professor Alkemade of Rijksuniverseit Utrecht in preparing the work on

signal-to-noise ratios. A special note of thanks for advice, support,

and encouragement is extended to Professor James D. Winefordner and the

members of the JDW research group.











TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS . . . . . . . . ... . . ii

ABSTRACT. . . . . . . . . . ....... v

CHAPTER

I INTRODUCTION . . . . . . . .. . . . 1

II SIGNAL-TO-NOISE RATIOS IN ANALYTICAL SPECTROMETRY. . . 4

Noise and Signal-to-Noise Expressions. . . . . . 4
Mathematical Treatment of Additive Noise ........ 6
D.C. Measurement in the Presence of Background
Shot Noise . . . . . . . . . . 17
D.C. Measurement in the Presence of Background
Flicker Noise . . . . . . . . . 18
Other Measurement Systems in the Presence of Background
Noise. .22
Mathematical Treatment of Multiplicative Noise . . .. 22
Assumptions . . . . . . . . . 25
General Expression for the Relative Variance ..... 26
D.C. Measurement with a Current Meter for
White Noise . . . . . . . . . 33
D.C. Measurement with an Integrator for White
Noise . . . . . . . . . . . 35
D.C. Measurement with an Integrator for Flicker
Noise . . . . . . . . . . .35
Signal-to-Noise Ratio Expressions in Emission and
Luminescence Spectrometry. .37
Expressions for S/N for Single Channel Detectors. 37
Sample Modulation . . . . . . . ... .41
Wavelength Modulation . . . . . . ... 42
Conclusions. . . . . . . . . ... ..... 42

III MOLECULAR LUMINESCENCE RADIANCE EXPRESSIONS ASSUMING
NARROW BAND EXCITATION . . . . . . . ... .53

Assumptions . . . . . . . . . . . 53
Steady State Two Level Molecule . . . . . .. 55
Steady State Three Level Molecule. . . . . .... .. 62
Limiting Cases of Steady State Excitation. ....... .. 69
Steady State Saturation Irradiance . . . . ... .71







Page

Nonsteady State Two Level Molecule . . . . . .. 72
Nonsteady State Three Level Molecule . . . . .. 73
Conclusions. . . . . . . . . .. .... .. 84

IV PULSED LASER TIME RESOLVED PHOSPHORIMETRY. ....... .. 88

Introduction . . . . . . . . . . . 88
External Heavy Atom Effect . . . . . . . . 91
Analytical Applications . . . . . . . 91
Theory. . . . . . . . . ... ..... .. 92
Experimental . . . . . . . . ... .. . . 94
Instrumentation . . . . . . . . ... 94
Instrumental Procedure. . . . . . . 115
Data Reduction. . . . . . . . . .. 117
Reagents. .............. . . .. 118
Results and Discussion . . . . . . . ... 119
External Heavy Atom Effect of Iodide, Silver, and
Thallous Ions . . . . . . . . . 119
Lifetimes and Limits of Detection for Several Drugs 152
Comparison of Excitation Sources. . . . . ... 160
Conclusions. . . . . . . . . . . . 170

APPENDIX COMPUTER PROGRAMS USED FOR LIFETIME CALCULATIONS . 172

LIST OF REFERENCES. . . . . . . . . ... . . 179

BIOGRAPHICAL SKETCH . . . . . ... . . . . . 186








Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy


OPTIMIZATION OF SIGNAL-TO-NOISE RATIOS IN ANALYTICAL SPECTROMETRY:
THE EXTERNAL HEAVY ATOM EFFECT IN PULSED LASER
TIME RESOLVED PHOSPHORIMETRY

By

Glenn D. Boutilier

December 1978

Chairman: James D. Winefordner
Major Department: Chemistry

A treatment of noise and siqnal-to-noise ratios of paired readings

is given for additive and multiplicative noise using the relation between

the autocorrelation function and the spectral noise power. For additive

noise the treatment is limited to cases where the background shows only

either shot noise or flicker noise. In the case of multiplicative noise

the treatment concerns cases of white noise or flicker noise causing

signal fluctuations.

Radiance expressions are developed for molecular luminescence in

terms of steady state and nonsteady state concentrations. The excitation

source is approximated as a narrow line source since its bandwidth is

assumed to be much narrower than the absorption profile. Limiting

radiance expressions are given for both low (conventional) and hiah

(laser) intensity sources. Saturation irradiances for the 2-level and

3-level molecular systems are also given.

A pulsed source time resolved phosphorimeter is described. A

nitrogen laser and a flashlamp pumped dye laser are used as excitation

sources and compared with respect to limits of detection for benzophenone,






quinine, and phenanthrene. The external heavy atom effect has been

studied using iodide, silver, and thallous ions as external heavy atom

perturbers in an ethanol and water solvent at 77 K. Phosphorescence

lifetimes and relative intensities for carbazole, phenanthrene, quinine,

7,8-benzoflavone, and thiopropazate are given and the mechanism of the

external heavy atom effect is discussed. Phosphorescence detection

limits for several drugs are reported.











CHAPTER I

INTRODUCTIONr



The measurement of signals in optical spectrometry is influenced by

the presence of spurious signals, or noise. Some types of noise may be

eliminated by proper use of measuring equipment as in the case of pickup

of 60 Hz from the alternating current (a.c.) electrical lines in the

environment. Some types of noise are fundamental to a given experiment,

and although they may not be entirely eliminated, it is often possible

to minimize them. The quantity of fundamental importance in analytical

spectrometry is the signal-to-noise (S/N) ratio.

Noise will be considered briefly from a fundamental point of view.

The S/I ratios for cases where the signal is from the analyte and the

noise due to the background (additive noise) and where the signal is

from the analyte and the noise is a process which affects the magnitude

of the signal multiplicativee noise) will be derived for several dif-

ferent measurement arrangements and optimization of S/N ratios will be

discussed. General signal expressions in analytical spectrometry will

be given along with S/N ratios for analytically important situations in

emission and luminescence spectrometry. The generally useful S/I ratio

expressions will be discussed with respect to analytical measurements.

Radiance expressions for atomic fluorescence excited by both high

and low intensity sources have been given for both steady state (1-4)

and nonsteady state (5) situations for two and three level atoms. The








intensity of saturation and excited state concentration expressions have

been given for gaseous and liquid molecular systems (6-11). Despite the

success of radiance expressions in predicting the variation in atomic

fluorescence radiance with source spectral irradiance, no similar

expressions have been developed for molecular luminescence spectrometry.

Killinger et al. (12) have elegantly treated the molecular absorption of

OH molecules in terms of the broadening processes (13) influencing the

electronic absorption transition. This treatment was not concerned with

steady state concentrations of levels or electronic molecular absorption

in general.

In atomic fluorescence expressions, it is often possible to assume

steady state conditions when using pulsed source excitation due to short

lifetimes. In flames, the observed lifetime may be 10-fold or more

smaller due to the concentration of quenchers in the flame. For molecules

in flames, this is also often the case, and it may also apply to

fluorescence in the condensed phase. It can not, however, apply to

molecules which exhibit phosphorescence in rigid media due to the long

lifetime of the triplet state compared to the pulse width of the ex-

citation source. For this case, nonsteady state expressions will be

given.

Phosphorescence is a luminescence process where radiation is emitted

from the triplet state of an organic molecule. Time resolution in

phosphorescence spectrometry makes use of the difference between the

phosphorescence lifetime of a given molecule and the lifetimes of other

sources of interference such as stray light, fluorescence, or phospho-

rescence from the solvent. Aaron and Winefordner (14) have reviewed the

available techniques in phosphorimetry along with their analytical





-3-


applications. Two of these, the external heavy atom effect and the use

of pulsed excitation sources will be studied here.

Pulsed sources offer several advantages over conventional sources in

phosphorimetry (15). Higher peak source irradiance may be obtained and

therefore increase the signal. Phosphors with shorter lifetimes may be

measured due to the rapid termination of the pulsed source. The S/IN

ratio may be improved by using a gated detector with a pulsed source.

The entire phosphorescence decay curve may be easily measured to check

for exponential decay. The highest source irradiance available is from

pulsed lasers. The construction of a pulsed source time resolved

phosphorimeter using two different pulsed lasers as excitation sources

will be described. This system will be applied to the measurement of

phosphorescence lifetimes. Limits of detection for several drugs will

also be reported and compared with results using conventional phos-

phorimetry.

The reported sensitivity of phosphorimetry has been increased by

the external heavy atom effect using iodide ion (16,17), silver ion (18),

and thallous ion (19) as external heavy atom perturbers. The effects of

these heavy atom perturbers on the phosphorescence signals and lifetimes

of carbazole, phenanthrene, quinine, 7,8-benzoflavone, and thiopropazate

will be reported. Limits of detection using these heavy atom perturbers

for these compounds and several drugs will be reported and compared

with limits of detection without heavy atom perturbers.












CHAPTER II

SIGNAL-TO-NOISE RATIOS IN ANALYTICAL SPECTROMETRY



Noise and Signal-to-Noise Expressions


The quantum nature of radiation causes fluctuations for which the

term shot noise is colloquial. Shot noise ultimately limits the maximum

precision to which a signal can be measured to a statistically pre-

dictable level. In addition to the statistically predictable shot noise,

additional scatter in the values of the measured signal occur due to

excess low-frequency (e.l.f.) noise. The most common case of such noise

has a noise power spectrum which is roughly inversely proportional to

frequency and is termed flicker noise or 1/f noise. The cause of these

noise sources may be found in the light sources, the absorbing medium,

the detectors, and the electronic measurement systems used in optical

spectrometry.

Calculations of shot noise in terms of standard deviations and

noise power spectra generally do not present difficulties. Problems do

arise when 1/f noise has to be taken into account, since the integral

describing the standard deviation diverges. An adequate description

can then be given when use is made of the auto-correlation function of

the noise signals and when paired readings are considered; this treatment

yields general expressions for the signal-to-noise (S/N) ratio. In-

serting the specific time response and frequency response of the









measuring system and the specific noise power spectrum, one obtains S/N

expressions in the various cases from which optimal values of the time

constants can be derived.

The study of noise (20-24) forms part of the discussion of errors

in analytical measurements. Errors may be divided into: (i) systematic

errors (25) which may arise from the measuring procedure itself and from

unwanted signals produced by background, stray light, detector offset,

etc. which can be corrected for by various methods, including blank sub-

traction, signal modulation, careful calibration, etc.; and (ii) random

errors or scatter which are a result of random variations with time of

physical quantities or parameters that affect the signal reading, called

noise.

The root mean square (r.m.s.)-value of a noise source and the signal-

to-noise ratio are useful parameters to describe figures of merit of ana-

lytical procedures (26). These important analytical figures of merit

are (i) the relative standard deviation which is the reciprocal of the

signal-to-noise ratio; (ii) the analytical limit of detection which is

the amount (or concentration) of analyte that can be detected with a

certain confidence level by a given analytical procedure; (iii) the

sensitivity of the analytical method, which corresponds to the slope of

the analytical calibration curve. The limit of detection is defined by

XL Xbl kibl
CL(or qL) S S (11.1)


which ties together two of the analytical figures of merit, namely the

limit of detection (concentration, CL, or amount, qL) and the sensi-

tivity. S. The limit of detection is also related to the blank noise

level, ,bl, resulting from 16 measurements of the blank where Xbl is








the average blank, obl is the standard deviation of the blank, and k is

a protection factor to give a desired confidence level (a value of

k = 3 is recommended which gives a 99.67% confidence level).



Mathematical Treatment of Additive Noise


Several concepts are fundamental to the mathematical treatment of

noise. Frequently, it is required to calculate the average of a function

g(X) v.here X is a random variable and a function of time, X(t). This may

be accomplished by using the probability density function, f(X,t), of X

which gives the probability that X has a value between X and X + 'X at

time t. If f(X,t) is independent of time, f(X,t) = f(X), then the

variable X is said to be stationary. It is assumed that f(X) is nor-

malized so that f f(X)dX = 1. Ensemble averaging of a function g(X)

is defined as g(X = f g(X)f(X)dX where the bar means ensemble averaging.

The spectral noise power (noise power per unit frequency interval)

in terms of current fluctuations for shot noise is given by


(Si)sh(f) S = 2e T (11.2)


where e is the elementary charge, C, and i. is the j-th component in
'J
the current, A. The spectral noise power considered as a function of

frequency, f, is called the noise spectrum. The units of S. are A2s and

bars denote average values.

Excess low-frequency noise has a noise power spectrum which in-

creases towards low frequencies and has a frequency dependence often

given by f-" where ., is close to unity (flicker noise). In spectrometry,

1/f noise is the most common and so will be the only one discussed in









detail. The frequency below which 1/f noise becomes important depends

on the noise source and the signal level and can vary from less than 1 Hz

to frequencies over 1000 Hz. This noise will be termed flicker noise

throughout this manuscript despite the use of this term for a variety of

other concepts. The cause of flicker noise is not well-known. Various

models for 1/f noise in electronics have been developed (22) but most

seem to have little relationship with spectrometric systems. The major

sources of flicker noise involve drift of light sources, analyte pro-

duction, and detection. The spectral noise power in terms of current

fluctuations for flicker noise is given by

2
(Si)fl(f) = T2 (11.3)
S flf j


where f is the frequency, Kf is a constant with dimensions unity which

describes the low-frequency stability of the noise source and i is as
-2
defined previously. We note that the flicker noise power varies as i

whereas the shot noise power varies as ij; the r.m.s.-value of the flicker

noise is thus proportional to the mean current (so called proportional

noise).

Apart from the noise components mentioned there may occur peaks in

the noise power spectrum which are, for example, due to oscillations in

the flame-burner system, such as vortex formation in the gas flows and

resonances in the tubings. They may extend to the audible frequency

range and are then called whistle noise. The noise power in such peaks

is also proportional to the square of the photocurrent, as in the case

of e.l.f. noise.

When combining noises of different origins into a total noise ex-

pression, the method of addition must be carefully considered. For








example, if two noises with r.m.s.-values oa and ob exist together, the

r.m.s.-value of the total noise, oT, is given by


0 = 2 + 02 + 2co a (1I.4a)
T a b a b


where c is a correlation coefficient; Icl ranges between Icl = 1, in the

case of complete statistical correlation, and c = 0 in the case that

both noises are completely uncorrelated. Statistical correlation may

exist when both noises have a common origin (e.g. fluctuations in the

flame temperature).

Because noise is a sequence of unpredictable events, it is impossible

to predict a future value based upon previous values. However, by means

of probability theory, it is possible to state the chance that a certain

process will be in a certain state at a certain time (20,22), yielding

a distribution of probabilities for the possible states. A well-known

distribution is the Poisson distribution. It is found when events occur

independently, e.g. in time, then the variance of n events occurring

in a time period of given length equals the mean value of n, found when

the measurement is repeated a large number of times:

2 -
var n = = n (II.4b)


where o is the standard deviation of n.

In this chapter,the emphasis is on the S/N ratio of a measurement,

which is the ratio of a signal to the standard deviation of the signal,

as measured in the readings of a meter or an integrator.

In order to be able to compare the signal-to-noise ratio obtained

with different types of noise and with different measuring procedures,

and to find optimum values of the various characteristic times, one may








with advantage make use of the relation between the auto-correlation

function and the spectral noise power involved.

The auto-correlation function of a continuously fluctuating signal

dx(t) is given by


!,X (T) dx(t)dx(t + r) (11.5)


where a bar denotes the average of a large number of values found at

different times t for constant time difference T. In the case of

fluctuations, one generally makes dx(t) = 0 by subtracting the average

value from the signal. For a signal based on a purely statistical

sequence of events (e.g. emission of photoelectrons in the case of a

photocurrent in an ideal photomultiplier tube, upon which falls a con-

stant light signal), x(Tr) differs from zero only for T = 0, i.e.,

i';x(T) = 0 for T / 0. The values of dx(t) at different times t are

completely uncorrelated and the auto-correlation function is simply a

delta-function at T = 0. This case is typical for shot noise. However,

other noise sources may have a different character; in the case of e.l.f.

noise, the values dx(t) and dx(t + T) do show a statistical correlation

also for large r, i.e., q.x(r) differs from zero also for T / 0.

Statistical correlation for T / 0 also occurs when shot noise is ampli-

fied and registered by an instrument that has a "memory," e.g. due to

the incorporation of an RC-filter.

To obtain an expression of the noise in the frequency domain, use

can be made of the Wiener-Khintchine theorem (22,27), which relates the

auto-correlation function to the spectral noise power Sx(f) through a

Fourier transformation:





-10-


S (f) = 4 f dx(t) dx(t + T) cos(rT) dr (11.6)
x 0


= 4 J lx(T) cos(WT) dT
0


and


Sx(r) = f Sx(f) cos(wr) df (11.7)
0

with w 2nf.

The Fourier transform of a delta-function, which describes qx(T)

for shot noise, is a constant. The transform shows that the shot noise

power is evenly distributed over a large (ideally infinite) range of

frequencies, because of which it is also called white noise.

When a noise signal is processed by a measuring system, its sta-

tistical properties will generally be changed. When a meter with time

constant Tc is used, this meter will, through its inertia, introduce a

correlation-in-time which makes the auto-correlation function of the

meter fluctuations due to the (originally) white noise differ from zero

also for T / 0. It also changes the auto-correlation function of the

e.l.f. noise; consequently, the related noise power spectra are also

changed. When an integrating measuring system is used, an analogous

effect occurs. For white noise, integrated over a time Ti, a correla-

tion will exist between the results of two integration when they are

taken less than T. seconds apart. When they are taken more than T.

seconds apart, the results are again strictly uncorrelated. For e.l.f.

noise, a similar reasoning holds, i.e., an extra correlation is intro-

duced in the noise signal when the integrator readings are taken less

than Ti seconds apart; when the readings are taken more than Ti seconds





-11-


apart, only the correlations in the original signal contribute to the

correlation in the readings.

To relate the standard deviation of the signal, which is needed for

the calculations of the signal-to-noise ratio, to the auto-correlation

function and the spectral noise power, we follow the procedure outlined

in reference (24).

When one works near the detection limit, which is set by the back-

ground fluctuations, one usually applies paired readings. The background,

which has been admitted to the measuring system during a time long com-

pared to the time constant of the system, is read just before the signal

to be measured is admitted at t = t Its value is subtracted from the

signal-plus-background reading made T seconds later; T is called the

sampling time. This difference, Ax, is taken to be the signal reading

corrected for background where


x = Xs+b(to + s) xb(to) (I.8)


Equation 11.8 can be rewritten as


A> = >s(to + Ts)+ [db (t + ) dxb(t )] (11.9)


where dxb(t) is the statistical fluctuation in the meter deflection or

integrator output due to the background alone. The signal-to-noise ratio

(S/N) is then the signal reading, x (t + T ), divided by the standard

deviation onx, in the difference of the background fluctuations occurring

T seconds apart (see Figure 1). We assume the noise in the signal to

be insignificant as compared to the background noise, and so


S-x x(t + T )
Xs(to s
(II.10 )
rl L:;x












Representation of Signal and Noise Measured with a Meter


al Signal Photocurrent, is, vs Time and
a2 Fluctuating Background Photocurrent, ib, vs Time.
b1 Meter Deflection for Signal, xs, vs Time and
b2 Meter Deflection for Background, xb, vs Time.


KEY TO SYMBOLS:

is = signal primary photocurrent
ib = background primary photocurrent
ib = average background photocurrent
xs = signal meter deflection
xb = background meter deflection
"b = average background deflection
t = time
to = sample producing signal introduced
TS = sampling time
rc = time constant of meter damped by RC-filter
Tr response time of meter deflection
dxb(to) = fluctuation in background deflection from xb at to
dxb(to+T) = fluctuation in background deflection from xb at to + rs


Figure 1.





-13-


' i i









C - --7 ---- -( -.- -- -

t L---
I ----








---.-a. t L - -aY
I
i








1 --








i, / .













i , -t-c *?"C .. --;,-




-14-


with

x = [dxb(t + s) db(to)]2 1/2 (II.11)

From Eq. 11.11, the variance a can be straightforwardly expressed

as

2 2
S dx t + r + dxb(t) 2dxb(t + Ts)dxb(t ) (11.12)


Because the background fluctuation is assumed to be stationary, each of

the first two terms in the right-hand side of the latter equation is
2
equal to ob which is the time-independent variance of dxb(t). From the
2
very definition of the auto-correlation function, o may be rewritten as
: X

x = 202 -2dxb(t +T )dxb(t) = 2[,x(0) p( )] (11.13)

where
(2 2 2
= t+ = dxb(to) -
x(0) dxb(to s) = d' = ob

and

,Ix(Tr ) dx (t Ts )dx (t )
x s b o s b o

2
To calculate o, the auto-correlation function is expressed in

terms of the spectral noise power S. (f) of the background current

fluctuations and in the characteristics of the measuring system, using

the Wiener-Khintchine theorem. Therefore, x( s) may be expressed as


'x( ) =- = S (f)cos(2nfT )df (11.14)
0

where

S (f) = S (f)IG(f)2 (11.15)

and G(f) is the frequency response of the (linear) measuring-readout





-15-


system. In other words, the spectral noise power of the meter fluctua-

tions is the product of the spectral noise power of the background

current fluctuations, S. and the squared absolute value of the fre-
12
quency response of the measuring system, IG(f)|2, including the ampli-

fication of the photomultiplier detector. Since noise power is a squared

quantity, one needs here the square of the absolute value of the

frequency response; phase-shifts and the associated complex form of the

frequency response do not enter in the calculation of noise signals.

Substituting Eq. 11.15 into Eq. 11.14 gives


x(T ) = f Si (f)JG(f) 2cos(2nfr )df (11.16)
0 b

2
Using Eq. 11.16, Eq. 11.13 for o. may be rewritten as
x

2 = 2 ) Si (f)IG(f) 2{1 cos(2nfT )}df (11.17)
Ax 0 b

2
because cos(2nfT ) = 1 for T = 0; a. is therefore a function of the
s s -x
sampling time T and as T -l 0 both ".-. and xs approach zero. It
2
should be noticed that the factor 1 -cos2nfT (= 2sin 2fr ) stems from
s s
the use of paired readings. The noise components having frequencies f

for which frT = 1, 2, 3, etc. are completely rejected.

The signal deflection, x (t + T ), due to a constant signal current

is that is instantaneously applied to the input at time to is


xs (t + s) = G is X(Ts) (11.18)


where G is the d.c. response of the detector plus measuring system, and

x(Ts) is the normalized time response of the system used (meter or

integrator), to a unit step function. Introducing the normalized





-16-


frequency response of the measuring system,


SG(f) G(f)
g(f) G G (11.19)

Equation 11.10 for the signal-to-noise ratio finally becomes


S ix(S )
S--- (11.20)
[2 J Si (f) g(f) 2{I cos(2nfT )Jdf]1/2
0 b

This equation is the general expression for the signal-to-noise ratio

with dominant background noise in the case of paired readings with a

d.c. measuring system (meter or integrator).

To optimize the S/NI ratio for specific situations, we have to in-

troduce in Eq. 11.20:

a. the background noise spectrum S. (white noise or flicker

noise);

b. the time response x(Ts) of the meter or the integrator used,

and the associated normalized frequency response g(f), and to

determine the dependence of the S/N thus found on the sampling

time T and the other time parameters.

It is assumed that the photon irradiance to be measured has been

converted to an electrical signal through the photocathode of a photo-

multiplier. All currents, i, refer to primary (or cathodic) currents or

count rates, respectively. An anodic current, ia, is related to the

cathodic current, ic, by


i = iG (11.21)
a c pm

where Gpm is the average gain of the photomultiplier. This expression





-17-


can be used if one wishes to convert final expressions for S/I1 to anodic

currents.



D.C. Measurement in the Presence of Background Shot Noise


In this case, a constant signal current i is assumed to be applied

to the input at t = t whereas the background current ib is assumed to

be continuously present. The step response of a meter damped by an PC-

filter (see Figure 1) or the normalized response of a meter when a

constant d.c. current is suddenly applied at t = to, is


x(Ts) = 1 exp(-TS/Tc) (for Ts 0) (11.22)


where the meter time constant T = PC. The response time of the meter

is defined as

T = 2nTT (11.23)


After a time T the meter has reached its final deflection within 0.2',.

The squared absolute value of the normalized frequency response of such

a meter is


g(f)12 = 2 = (11.24)
1 + (2nT f) 1 + (f r )

Inserting Eqs. 11.22, 11.23, and 11.24 in Eq. 11.20, with S. (f) for shot
'b
noise, one obtains

S i (1 exp(-2nr /T )}
5S r' (11.25)
IN S (1 cos(2nfTs)}
f 2 2
0 1 + f /2
r


The integral in Eq. 11.25 can be evaluated by using





-18-


2
Ssin x 1 -2n
S-n 2 dx = (1 e )
0 n' + x

which yields


S is1 exp(-2nT /Tr)}1/2
I ( T(nS /T r)/2


For fixed Tr, the maximum value of S/N is reached for T = and is

i i
S s s 1/2
N S 1/2 (2eT 11/2 r
(S o/Tr) (2bei


11.26)


11.27)


Since the value is reached within 0.2% for T = T the sampling time T

can be restricted to that value. A larger value of Ts is only a waste

of time; a smaller value yields a smaller S/IN ratio. Equation 11.27 shows

that the S/N ratio is proportional to the square root of Tr and thus

improves with increasing response time T provided T T r



D.C. Measurements in the Presence of Background Flicker Noise


Substitution of the spectral noise Si (f) = Ki b/f into Eq. 11.20
yields
yields


S : is[1 exp(-2nTs/T )]
N _-9 l COS(27fT ) 1/9
{2K-i 2 df}
0 f(1 + f2T )


(11.28)


This expression is valid for any Ts and Tr, but can be evaluated only by

numerical methods. It is possible to simplify this expression by intro-

ducing two new variables with dimension unity. Let B and z be defined

as


























































C
0
c0r
4-J
U


LL






0




CL.

U
r-
E






0
-J
*I--
r-







C-
C,





.*.--

0)


L.







L20U





-21-


E 2n T / T /'c (11.29)


z 2nfr (11.30)


Substituting these new variables into Eq. 11.28 leads to

i exp(-e ) )
S s
-- (11.31)
2 2 1 cos z 1/2
2Kfib J 2 2dz
0 z(l + z / )

or

i
S s f(:) (11.32)
i 2-2 1/2 ( 32)
(2fKib

where

f(A ) = [1 exp(- )] (11.33)
1 cos z dz 1/2
6 z(l + z 2/2)

Numerical evaluation (24) of f(e) gives a maximum of approximately 0.88

at ~ approximately equal to 0.8, i.e., r~ :: T /8 or T s1 0.8 T and

f(8) falls to zero as E tends toward zero or infinity. A plot of f(Q)

vs t is given in Figure 2.

The important point is thus that the maximum S/I for flicker noise

is dependent only on the ratio /r and not on rs and Tr individually, and

so there is no gain in S/N here when we make T (r 8T ) larger. Evidently

in the flicker noise limited case, the increased smoothing effect of a

longer time constant r = 2nr is just offset by the increase in low-

frequency noise from the equally longer sampling time Ts, due to the

1/f-dependence of the flicker noise power spectrum. One can also show

that for a noise power proportional to f-' with a > 1, the S/N ratio

even decreases when Tr (and T ) is increased.
r s





-22-


The optimum S/Nr for background flicker noise is therefore

1 1
S. s s (11.34)
1 2-2 1/2-
(2.6K ibI 2cdmib


where tdm = 0.81 Kf is defined as the flicker factor for paired d.c.

measurements.



Other Measurement Systems in the Presence of Background Noise


Many other measurement systems may be used in analytical spectrometry

other than d.c. meter systems. Other d.c. systems possible are d.c.

integration, photon counting with a rate meter, and photon counting with

a digital counter (digital equivalent of integration). Modulated, or

a.c., systems such as lock-in amplifiers or synchronous photon counting,

may be used with meter (current or rate) and integration (counter)

output. Detailed derivation of the S/N ratio expressions for background

shot and flicker noise has been given (28), so only the final expressions

for the S/N ratio will be given here. In the a.c. cases, it is assumed

that the signal is modulated at frequency f mod while the background

signal is not modulated. In Table I, the S/N ratios are given for the

different measurement systems discussed for background flicker or back-

ground shot noise. In Table II, the flicker factors, ,, are given for the

different measurement approaches.



Mathematical Treatment of Multiplicative Noise


In the discussion of additive noise, it was assumed that fluctuations

in the meter deflection due to a fluctuating background constituted a




























-

- [ 0


S-









LI
LV)











-r--
-0
C
'V














Ln
a)




- E


















(
4--
0














-n
C



O

*-




vi


0
L




-0r


SE

W-

0v
'4-

Ca

01


I 0a
-u5
(V 0









-.
I-


U

10
















U

-D





















































E
a)

>1
0







E

0a
(U
a)


0

1.


((V


LL


U .-

I- c\j

LL.















4-' j





















(1
4-

4-)
C



C

5-
L
l_}


|. I i-
[ -- 10
A1 |1 C









Ql:







5-


















0
Icr


















(V


En=
Cr'I













O
En.u


L



0
C





-r-

-r-


I 5-


IA


-23-


Ln
I ;

UL-


4-,
c-


u
OE


U


O *
C)

3 -0 .
0 (0 E








U U O"












*r" 3
EoI
.0 -01
Q: 41' C:










0










r o
a)---


I- --
L .













O
U O .
-0
-CO
0 0





Cn:' a
E o
-r-


SC C-
5.., U 1
L -0-0




LO
EC -







S*-> -O
C U




u -- 3
C 0


U
r

C ui




U 1
<5 L.-L





1 *






e-- h


-I


[ ..

Li5-








1- |.-
1=



































t-


IA





S-
4-




CU

5-
L
1_)


.0


C\j















LL











vi








I-

1-




5-





-24-


Expressions for Flicker Factors, r, for Several Measurement
Approaches


Measurement Device d.c. a.c.


Current Meter




Integrator


am f(7/2fmod)1/2




ai Kf/(2fmod /2
"'ai f mod


m .65Kf = 0.81 Kf

4di ~2 = f083 Kf



di = 0.83 K,


Synchronous Counter


Table II.


i = K f(n2/f )od12
'si f mod





-25-


stationary fluctuation process. The background current, ib, was assumed

to have been applied to the meter for a long time before a reading was

taken. In the case of multiplicative noise, noise is introduced

simultaneously with a signal due to the analyte. If one applies paired

measurements such as the measurement of a reference (standard) followed

by measurement of an analyte signal, the very nature of the noise source

considered makes it impossible to ignore the noise in one of the measure-

ments. Since these signals are read after a sampling time T which may

be shorter than the response time, Tr, a stationary state of the meter

deflection may neither be reached for the average signal nor for the

fluctuations inherent to the signal. It is necessary to deal with the

transient response of the meter to fluctuations.



Assumptions


The assumptions used in this model of multiplicative noise are

(see Figure 3):

(i) The input analytical signal, is(t), and reference signal,

ir(t),are noise-free;

(ii) the time dependence of the input signal is a step function,

is(t) = is for T < t < T + Ts,

i (t) = i for 0 < t < T and

is(t) = i r(t) = 0 for t outside the given intervals;
(iii) at t = 0 and t = T, the meter deflection caused by the

preceding signal has decayed (T T ) or been reset to

zero;

(iv) no additive noises are present;




-26-


(v) is is proportional to the analyte sample concentration (CS)

and ir is proportional to a reference parameter (Cr) which

may be a calibration standard, excitation source intensity

in luminescence spectrometry, etc;

(vi) a "multiplication factor," G(t), is a stationary, Gaussian

noise process which produces multiplicative noise and is

given by G(t) = G + dG(t);

(vii) after "multiplication," the input signal i(t) is transformed

into the multiplied signal A(t) where A(t) = G(t)i(t);

(viii) the meter deflection x(t) and A(t) are related by

A(t) = + x (11.35)
t) dt T

(ix) the estimate of the analyte concentration, Cs, is given by

x (T + T )
Cs Ts S C (11.36)
5 xr T s) r


Several points should be carefully noted. The noise in the multiplica-

tive factor, G(t), is itself a stationary noise process, but x(t) is

not a stationary noise process. The reference signal, ir, and the

reference parameter, Cr, have been defined in a completely general

way. The most common case in analytical spectrometry is that the

reference is a standard of known analyte concentration. It is possible

that other references may be used, such as an internal standard.


General Expression for the Relative Variance

From Eq. 11.36, the differential of C may be written as

dC dx (T + Ts) dr(T)
s (T + (11.37)
C xs(T + ) xr(7 )
s s 5 r
































ro
L









4--





Cr






C 40 -L



U U >,
) C a 0 rO0
Q)J 4 o -'




C OO
0 0




a)-- U--- > O
L.) 4) Q ,J 4- L.-
-- I+ 0 n -



S0) -- 0--
S 4--, 0


> 00
CUi >0

=3 Q) n 4-' a)L

-0 _0 COU 0 "




U U U L E rC--- rJ-
QJ ) U -r- r 4---
O --- WU L- U EC
0 ( u v,*- u E
S*J O; U 0 C O *.- O
C L1 " + Ca C: 0 C -- ~ -
0 --J c Q -j- "r U CM c- Lr,-
- L L La Q) .-, -- S-- m r
-* 1 Q Lo W W W 4- SJ J -- *
ro 4-)L M- EEE E--W >
4 (1) () C) *r- a -.-- 3 > (1
c V E E E U 1 4- E r. -'
a)

a) " 1. -" C "I"0
-4-) A -


x -L- L 0 0U
X X ,-- --- "O





L.

r.-j




-28-


4-
I I


11






I--








N---- 4 c
( S ^ v-
() 5- ;


S ^ "


< a.





-29-


2
and the variance of Cs, oC is given by
s

2 = ddxs(T + T ) dx (s )2 ,
C x (T + T ) x (Ts CO (11.38)


The relative variance of CS may be written as

2
C (T )2 dx (T)dx (T + T
s2 --_ -r s (11.39)
C Y2 dx( 2 x (T + T ) r (T)
r s
s Xrs) s s r

where use has been made of the fact that (dx /xs)2 (dx /xr) 2 The

S/N ratio is given by Cs/os. We wish to find how the S/H ratio depends

on T T and T for given statistical properties of dG(t) and what the

optimum measurement conditions are.

From the definition of A(t) and G(t) and integration of Eq. II.35,

the expression for x(T) is


X(sT = ircG[l exp(-Ts/T )] (11.40)

For general expressions, the subscripts s and r will be dropped. For

A(t),an arbitrary function of t for t > 0 and zero for t < 0, the

general solution of Eq. 11.35 is
T
x(r ) = x(0)exp(-T /T ) +exp(-T s/T ) f exp(u/T )A(u)du (11.41)

where u is a dummy integration variable (23). Treating the meter de-

flection from the reference signal, x r (), and using the definitions

of A(t) and G(t) with x(0) = 0, it follows from Eq. 11.41 that


Xr(T ) = irTcG[I exp(-T s/T )] +

r rc 0 c
irexp(-T-/Tc) ? exp(u/Tc)dG(u)du (11.42)
0





-30-


or (see Eq. 11.40)


X (rT) = X ) + d r (Ts) (11.43)

and dx r( ) is given by
Ts
dxr(T) = i exp(-Ts/ c) I exp(u/T )dG(u)du (11.44)
0

From the previous evaluation of x (T ), the expression for the meter

deflection due to the analyte signal is x (T + T)= xs(T + T) + dx (T + T )

where


x (T + T ) = TG[1 exp(-T /T )] (11.45)


and
T+Ts
s
dx (T + T ) = i exp(-TS/Tc) I exp[(v- T)/T ]dG(v)dv (11.46)
T

where v is a dummy variable for integration.

To find the expression for dx (T )dx (T + T ), Eq. 11.44 and

Eq. 11.46 are multiplied and ensemble averaged. It is found that


dxr (s)dx (T + T ) iri exp(-2r c )

T T+T
S- du f exp[(u+v- T)/ c]dG(u)dG(v)dv (11.47)
0 T

The ensemble average over a double integral may be replaced by a double

integral over an ensemble average. Equation 11.47 can be rewritten as


dxr(T )dx (T + TS) =i i exp(-2T /T ).
r s rs 5 rs sc

Ts T+Ts
du f dv exp[(u+v- T)/T ]dG(u)dG(v) (11.48)
0 T





-31-


Because dG(t) has been defined as a stationary noise process, it

is possible to define the time-independent auto-correlation function

of dG(t) by


>G(s) = dG(t)dG(t + s) ( 1.49)

The term dG(u)dG(v) is therefore equal to 'G(v u). Rearranging Eq. 11.48

and replacing the integration over v by y = v u for given u results in


dx (r )dx (T + T ) = i i exp(-2T /T )-
r s s s rs s c

T T-u+Ts
f du exp[(2u- T)/Tr ] f exp(y/Tc)G(y)dy (11.50)
0 c T-u

This is the general expression for dx (Ts)dx (T + T )

In an entirely analogous fashion to that in which the expression

for dx s )dxs(T + T ) was obtained, the expression for dx (T)2 is found

to be

2 2
dx (Ts) = i exp(-2TS/TC)
r S r S C

s -U+T
*f du exp(2u/Tc) f exp(s/ ),G(s)ds (11.51)
0 -u

where s = u' u for constant u.

Substituting Eqs. II.40, 11.50, and 11.51 into Eq. 11.39, the ex-

pression for the relative variance of C is

2
C 2exp(-2i /T ) s -u+T
S 2-2r fc 2 duexp(2u/Tc) f exp(s/T),G(s)ds +
Cs Tc2[I exp(-T/T) 0 b -u

's T-u+T
f du exp[(2u-T)/lr] f s dy exp(y/ ),G (y)J (11.52)
0 T-u




-32-


The integral over u may be factored out, the integration variable y

replaced by s = y T, and the integrals over s combined. This results

in
-2 Ts

s 0
C2 T22[I exp(-T /T )]2
s c S c

i-U+T
f* s ds exp(s/Tc [ (s) iG(s + T)] (11.53)
-u

From Eq. 11.14, the Wiener-Khinchine theorem,


q.G(s) 4,G(s + T)= 2f SG(f)sinnf(2s + T)sinnfTdf (11.54)
0

Substituting Eq. II.54 into 11.53, gives the final, general expression for

the relative variance of Cs, which is Eq. 11.55.

2 rs
oC 4exp(-2T /T') f du exp(2u/Tr)

C TC[1 exp(-T /T)]2

-U+T
S c s C


Ss ds exp(s/T )
-u

f SG(f)sin7f(2s + T)sinnfTdf (11.55)
0

The integral over u is defined over the range 0 s u < T s T, and the

integral over s is defined over the range -u s s -u+T.

Up until this point, the derivation of the expression for the rela-

tive variance of Cs was general for SG(f), T Tc, and T subject to the

constraints of the assumptions. The divergency of flicker noise as

f 0 is neutralized by the two sine functions of frequency, f, in





-33-


Eq. 11.55. For mathematical evaluation, the order of integration in

Eq. 11.55 may be reversed. As is usually the case, it is complex to

evaluate.


D.C. Measurement with a Current Meter for White Noise

A case of interest is the case of a white noise spectrum. It is

possible to define a correlation time, TG, of noise dG(t) by

Gf +(s)ds
0
TG = 0 ) (11.56)


where ,iG(0) = dG(t). Because ',G(s) differs from zero only for Is ,: TG'

while TG Ts' c, and T, for this case, SG(f) is a constant over the
-1
relative frequency range, but falls off at 2nf TG. Starting from

Eq. 11.53, +G(s + T) = 0 because (s + T) nn TG. Because IjG(s) exists

only for s 0, the integral over s can be approximated by f IG(s)ds.

It is a valid approximation as 0 u :< T ; s is within the integration

limits of -u and -u + rs. From Eq. 11.56, the definition of iG(s), and

the approximation of the integral over s, Eq. 11.53 becomes
2 T2
"C 2exp(-2Ts Tc) exp(2u/Tc)du dG TG
5 0
C 2[1 exp(-T /)]1157)
CS Tc eXP(-TS T


Making the substitution z = 2u/c and evaluating Eq. 11.57 gives

2 2
Cs 2 dG' TG[1 exp(-2T s/Tc)]
(11.58)
2 ( 58 )
C G'c [ exp(-T /T )]2

From the definition, dG' = 'G(0), the inverse Wiener-Kinchcine theorem,





-34-


and Eq. 11.56 (see Eq. 11.6)


dG = SG(0)/4rG (11.59)


Substituting Eq. 11.59 into Eq. 11.58 yields

2
OCs SG(0)[1 exp(-2 s/T )]
2 )--2 (II.60)
C2 2G rc[1 exp(-Ts/ )]


The S/N ratio is therefore


S G,2hc [1 exp(-T /T )]
S c s c (11c61)
(II.61)
:SG(0)[1 exp(-2T/T c)]


The S/N ratio is found to be independent of T, or in other words,

the S/N ratio is unaffected by the time between measurement of the

reference signal and the analyte signal. The S/N ratio is maximum when

T -+ *. In practical measurements, the maximum S/N ratio is obtained

when T = 2nTc where 2nTC has been defined as the response time, T in

Eq. 11.24. In terms of the response time, the maximum signal-to-noise

is given by


S r
max = (11.62)
N SG(0)7-

If this equation is compared with Eq. 11.27 for the case of background

shot noise, it is seen that the S/N ratio increases in both with VrT.
r
It should be noted that the expression for shot noise may not be sub-

stituted here for SG(0) because shot noise is not a multiplicative noise.

All that can be specified is that for the white noise case SG(0) is

constant. The S/N ratio will also increase as GSG0O) decreases.





-35-


D.C. Measurement with an Integrator for White Ioise


The case of an integrator may be derived from Eq. 11.61 by taking

the limit as c '" for an integration time T. = r (28). The result

for the case of white noise is given by


,G 77
S i (11.63)


This shows an improvement in S/l ratio over a d.c. meter by a factor of

,n assuming T. = T
1 r


D.C. Measurement with an Integrator for Flicker Ioise


It is necessary to assume that ri *. rc, as was the case for the

integrator in the case of white noise. Starting from Eq. 11.55,

setting SG(f) = 2K/f for flicker noise, and approximating exp(2u/Tc),

exp(-2s /rc), and exp(s/Tc) by unity give


2
oC 4YK2
f f 1_l (_ T 2
s 2 (1- 1)2n(T T) + -L + 1)2(T + T -
C G i i


(T )2 nT ?nT. (11.64)
Ti


With a fixed integration time T., the minimum value of T is given

by Tmin = Ti (see assumptions). Solving for the S/N ratio gives


S (T= T ) G (11.65)
2 1 2Kf.v n4


If T is increased relative to Ti, for the limit of T >> Ti, the





-36-


signal-to-noise ratio becomes


S G
(T >> i) = (11.66)
2Kf '2 +fn(T/ .i)


As T increases, the S/N ratio decreases. For a fixed total measure-

ment time, the optimum S/N will be achieved by making n measurements of

reference and standard with T = T. and averaging the results, which

increases the S/N ratio by a factor of 'n. This conclusion has been

reached by Snelleman (29) and Leger et al. (30) for the case of additive

flicker noise. In practice, there is a fundamental limit to the amount

of improvement that may be achieved by this procedure. In the model for

multiplicative noise, only multiplicative noise sources have been treated.

All signals in analytical spectrometry will have shot noise, and if the

integration time becomes short enough, the shot noise may become the

dominant noise source. In this case, there will be no improvement in

S/N ratio as n is increased. For the case of multiplicative white noise,

there will be no difference between making one set of paired measure-

ments of sample and reference or n sets during the total measurement

time. The general conclusion is that the optimum signal-to-noise ratio

will be achieved when the sample and reference pair are measured as

rapidly as possible during the measurement time.

It is not possible to evaluate the case of a current meter for

arbitrary T c T and T without numerical integration. If one assumes

T >> Tr, then the noise can be treated as "quasi-stationary." In this

case,the conclusions for background flicker noise should apply. Again,

it is optimal to make several measurements and average the results,

which is the same conclusion reached for integration.





-37-


Signal-to-Nroise Ratio Expressions in Emission and
Luminescence Spectrometry


Expressions for S/N for Single Channel Detectors


It should be emphasized that in the previous discussion only one

noise source was considered in calculating the signal-to-noise ratios.

However, when making measurements in analytical spectrometry, more than

one noise source occurs and so must be considered whatever measurement

system is being utilized for the signal measurement. In this section,

only emission (atomic and molecular) and luminescence (atomic and

molecular) spectrometry will be explicitly considered. No attempt will

be made here to give general expressions for absorption (atomic and

molecular) spectrometry, although the expressions for emission and

luminescence spectrometry can be applied, with some changes, to absorp-

tion spectrometry, which is somewhat more complex due to the necessity

of making ratio measurements and the nonlinearity of absorbance with

analyte concentration. The noises occurring in emission and luminescence

spectrometry will be explicitly discussed and evaluated in this section,

particularly with regard to how the noises combine to give the total

noise in the measurement.

In general, shot noises are simple to consider since they add

quadratically, i.e., no correlation between these noises. Flicker

noises are much more complicated to handle because they may be depen-

dent, independent, or a combination of dependency and independency.

Although high frequency proportional noises are similar in complexity

to flicker noises, they can be omitted in the following treatment because

such noises can be minimized by proper selection of the frequency of the





-38-


measurement system. In the following treatment, flicker noises will be

assumed to be completely dependent or completely independent (no cor-

relation coefficients) according to the best experimental evidence

available to the authors (31-33). Although the most general expressions

should contain flicker noises with correlation coefficients, such ex-

pressions would be exceedingly complex and of little use since correla-

tion coefficients for flicker noises are rarely available. It was

necessary in the present treatment to assume the linear addition of

analyte emission or luminescence flicker noises to the related "back-

ground" flicker noises (background emission in emission spectrometry and

source related background, such as scatter and luminescence background

in luminescence spectrometry); this addition is not exact because analyte

flicker occurs only during the sample and not the blank. Nevertheless,

the expressions to be given should be good estimates of S/N for actual

experimental situations. Finally, tables of expressions and evaluations

of parameters will be utilized where feasible to simplify the expressions

and evaluations of the expressions. The S/N expressions to be given will

contain various parameters, such as total measurement time and counting

rates, which are evaluated according to the analytical system under

study, flicker factors which are evaluated according to the analytical

system under study and the measurement method, and constant terms

characteristic of the measurement method.

General S/N expressions (digital case only) for atomic or molecular

emission spectrometry and for atomic or molecular luminescence spec-

trometry, are given in Table III. All terms are defined at the end of

the table. The power terms, p, q, r, u, and w, are also evaluated in

Table III for the cases of CW (continuous excitation-continuous emission





-39-


or luminescence and continuous measurement), AM (amplitude modulation

of emitting radiation in emission spectrometry or of exciting source in

luminescence spectrometry), WlF (wavelength modulation of optical system

to produce an a.c. current for the analyte), SM (sample-blank modulation,

i.e., repetitive measurement of sample and blank), Al + WM (double

modulation where the optical system is slowly wavelength modulated while

rapidly amplitude modulating the signal as described above), and AM + SM

(double modulation where the sample and blank are repetitively and

slowly introduced while the amplitude is rapidly modulated as described

above). Other double modulation approaches, as UJ1 + SM, and triple

modulation, as AM + WM + SM, result in little gain in analytical figures

of merit and are more complex and so will not be discussed here.

Modulation methods are only useful in minimizing flicker noises

(any noise source which is present during both halves of the modulation

is reduced since is given by the appropriate AC-expression, i.e.,

ai for the synchronous counter, rather than by the d.c. integrator
expression, (Ti > .)
di di ai
In Table IV, the appropriate flicker factor, ,di or (ai for the

d.c. integrator or digital synchronous counter, respectively, is noted.

In Table V, evaluation of the duty factors for the various measurement

modes and for the various duty factors in the general noise expressions

defined in Table III (at end of table) are given. The duty factor is

generally defined as the fractional on time for any given process by any

type of measurement mode.

The expressions in Table III with the definition and evaluation of

terms in Tables III, IV, and V describe all measurement modes in emission

and luminescence spectrometry except for those cases where the emission




-40-


source in emission spectrometry or the excitation source in luminescence

spectrometry is pulsed and the detector-electronics system is gated with

or without time delay between the termination of excitation and the

initiation of measurement (31). In Table VI, expressions for duty

factors to describe source pulsing-detector gating are given with

definition of terms. The duty factors, DEN and DL replace the values

of 1/2 or 1 in Table V for CW, AM, WM, SM, AM + WM, and AM + SM measure-

ment modes. The CW mode for source pulsing-detector gating implies that

a blank is determined in order to correct for background, interferent,

and dark counts in emission and for background, interferent, scatter,

and dark counts in luminescence. The AM mode for source pulsing-detector

gating implies that a blank is determined as above for the CU mode but

also in between source pulses for a time period of t s, dark counts

are observed in emission spectrometry and dark counts, analyte emission,

and background emission are observed in luminescence spectrometry. The

other modes have not been used for analytical emission and luminescence

spectrometry but would involve the following: WM mode means that every

other pulse is "on" wavelength and alternate pulses are "off" wavelength

in either emission or luminescence spectrometry--again a blank must be

"run"; SM mode means that one or more pulses occur for the sample and

one or more (the same number as for the sample) occur for the blank and

then the process is repeated for either emission or luminescence

spectrometry--in this case, in luminescence spectrometry, a separate

source of measurement must be "run" to determine the emission signal;

double modulation methods, AM + WlJr and AM + SM are of interest only for

luminescence spectrometry and involve a combination of the above modes.

Therefore, to obtain the appropriate S/N expression, one takes the




-41-


appropriate expression from Table I with noise terms described by the

expressions at the end of the table; the flicker factors are those

listed in Table II. The duty factors, except for DEM, DLM', and DGD

are those in Table V, and the ones for DE', DLM, and DGD are given in

Table I11.



Sample Modulation


Sample modulation, SM, was discussed in the previous section

However, this rather unique approach to analysis (34,35) requires some

specific comments. In SMi, the sample and blank are repetitively measured

for n equal time periods each, and so unmodulated flicker noise sources,

e.g., flame background in atomic fluorescence flame spectrometry, con-

tinuum scatter or molecular band interferents in atomic fluorescence

flame spectrometry, etc., will be reduced as the modulation frequency,

f mod' increases and the measurement system's noise bandwidth, Lf,

decreases, i.e., the flicker factor, .i is related to zf/fmod by

1
a A'f 2nto = 1 (11.67)
ai f 1 n
F mod 2t
0

where to is the observation time of sample or blank per cycle and n is

the number of sample-blank cycles. As the number of sample-blank cycles,

n, increases (ai decreases inversely with ,'n. There is a practical

limit to fmod and therefore to Af/fmod' namely, the time to mechanically

change from sample to blank with no memory effects, and so fmod < 10 Hz,

which may not be as effective in removing noise as WM modulation which

requires twice the number of measurements. In addition in SM, an





-42-


"ideal" blank, (contains everything in the sample except the analyte)

must be prepared and used.



Wavelength Modulation


In WM, all flicker noise sources which are present "on" and "off"

the analyte measurement wavelength are reduced, i.e., ; < d. If
ai 'di
the samples and standards are identical in all respects, except for the

analyte, then WJM corrects the signal level for unmodulated signal com-

ponents and reduces flicker noises due to these sources. Because 11WM can

involve the mechanical movement of a small refractor plate or mirror in

the optical train of a spectrometer, it is possible to obtain higher

modulation frequencies, e.g., < 100 Hz, than in SM (but lower than in

AM); therefore, because i. c ",f/f as in Eq. (11.67), can be made
Sai c mod ai
smaller than for the corresponding noise in SM. Of course, in luminescence

spectrometry, any analyte emission signals must be corrected for by a

separate "source off" measurement unless the sum of emission plus

fluorescence is desired. If line interferents are present, WM may

result in an erroneous analyte signal, whereas in SM, assuming the line

interferent is present in sample and blank, the analyte signal level

will be correct but the noise is still degraded.



Conclusions


The major conclusions which can be drawn from the treatment of

signal-to-noise ratios are

(i) For the cases of white noise, whether additive or multiplica-

tive, the S/N ratio increases as the square root of the





-43-


response time, T or the integration time, T., for current
r 1
meters and integrators respectively;

(ii) For background shot noise limited cases, modulation techniques

will give S/N ratios '2 times poorer. Sample modulation is

an exception, because it is necessary to measure the blank

regardless;

(iii) For the cases of white noise, whether additive or multiplica-

tive, the S/N ratio is independent of the rate at which sample

and background or sample and reference are measured;

(iv) For the cases of flicker noise, whether additive or multiplica-

tive, the S/N ratio is approximately independent of response

time or integration time;

(v) For the cases of flicker noise, whether additive or multiplica-

tive, the S/N will decrease with increasing sampling time

relative to a fixed response time. It is optimum to ma'e the

integration or response time as short as is practical and

repeat the pair of measurements n times;

(vi) The case of multiple sampling during the measurement time for

background flicker noise cases is essentially the same as using

an a.c. system where the signal is modulated and the noise is

not modulated;

(vii) If both the signal and background noise are modulated in a

background flicker noise case, no increase in S/N ratio

results;

(viii) In a background flicker noise case when using an a.c. system,

it is optimum to make af/fmod as small as possible (either

with small if or large fmod);
mod





-44-


(ix) The optimum system in the case of multiplicative flicker noise

is to measure sample and reference simultaneously. The best

reference in most cases is a calibration standard, but it is

often impossible to measure a signal and a standard simul-

taneously. In some situations,an internal standard, excita-

tion source intensity, etc., measurement may be made simul-

taneously and will improve the S/N if the source of multiplica-

tive noise affects both in the same way and is the limiting

source of noise. An example is that taking the ratio of the

signal to the excitation source intensity in luminescence

spectrometry will not improve the S/N ratio if the major

source of multiplicative noise is connected with the sample

introduction system.





-45-


Table III.


General Signal-to-Nloise Ratio Expressions for Emission and
Luminescence Spectrometry with Definition of Terms


E rS + n2
E BES


S + 2IDS+ (EF +
IS e DS, E


2 ql eF + 2A BF)2 + (2 WNDF 2 + (2 1A) 2
e


Measurement Mode q w

CW 1 1
Atl 1/2 1/2
WM1 1/2 1/2
Sri 0 1/2


L 2
LS


2 + 2 +
ES I f
f'


2 + rI2 + rt 2 ( + 2u F +2 i ) 2
SS S DS (LF +2 F S


2P(N~i F + 2 q1 l )2 +
E F


(2 1DF)2 +


Measurement nlode p q r u w

C 1 1 I 1
AM 0 1/2 1 1 1/2
Wi 1 1/2 1/2 (continuum) 1/2 1/2
1 (line)
SM 1 0 0 0 1/2
AM + WM 0 0 1/2 (continuum) 1 1/2
S1 (line)
AM + SM 1 0 0 0 0


(2 lA 1/ 2




-46-


Table III. (continued)


Definition of Terms

PrES = analyte emission shot noise = DE DMR t m, counts

NBS = background emission shot noise = DEDoRBt m, counts

NI S = interferent (in matrix emission) shot noise = DENR I tm, counts
e e
NDS = detector dark shot noise = DGDRDtm counts

NLS = analyte luminescence shot noise = DL IDWMDORLt counts

1SS = scatter (source) shot noise = /DLM D.DoR t counts

N, S = interferent (in sample/blank) luminescence shot noise =

DLMDRI tm, counts

2"wA = amplifier readout noise (generally negligible in S/N measure-

ments), counts

NEF = analyte emission flicker noise = EFDEMD DOREtm, counts

2q BF = background emission flicker noise = 2qBFDD DSBDoRB m, counts

2 qI F = interferent (in emission flicker noise = 2q 1 F DEr.I tm, counts
e e e
2 NJDF =detector dark flicker noise = 2wDFDGDRDt counts

2r SF = scatter (source) flicker noise = 2rSFDF M DI.IDSBD m, counts

2ul I F = interferent (in sample/blank) luminescence flicker noise =

2u FDLMDOR Iftm, counts
TLF = analyte luminescence flicker noise = 2 LDL BD R t counts
LF 2L L4 l jSB L O m
S = analyte emission signal = Rt counts

SL = analyte luminescence signal = DL DDRLt, counts
tm = measurement time for one spectral component, s (see Figure 3

and text)

DLM = amplitude modulation factor for luminescence spectrometry,

dimensionless




-47-


Table III. (continued)


Definition of Terms (continued)

DEn = emission modulation factor for emission spectrometry, dimen-

sionless

DS = sample-blank factor, fraction of time sample is "on," dimen-

sionless

D = wavelength modulation factor, dimensionless

D,1 = wavelength modulation factor for narrow line, dimensionless

DO = factor for correction for emission in luminescence spectrometry,

fraction of time emission or luminescence (equal times) is

measured, dimensionless

DGD = gated detector factor to account for fraction of time detector

is gated "on," dimensionless

RE = photoelectron counting rate of analyte emission, s

RB = photoelectron counting rate of background emission, s-

R = photoelectron counting rate of interferent in emission spec-
e -1
trometry, assumed to be in both blank and sample, s

RS = photoelectron counting rate of source scatter in luminescence

spectrometry, s1

R = photoelectron counting rate of interferent luminescence in
-1
luminescence spectrometry, assumed to be in sample and blank, s

RD = detector dark counting rate of detector, s
-1
RL = photoelectron counting rate of analyte luminescence, s-

EF = flicker factor for analyte emission flicker, dimensionless

S= flicker factor for emission interferent flicker factor,
I F
e
dimensionless




-48-


Table III. (continued)


Definition of Terms (continued)

.BF = flicker factor for background emission flicker factor,
dimensionless

SF = flicker factor for source scatter (in luminescence spectrometry)
flicker factor, dimensionless

F = flicker factor for luminescence interferent (in luminescence
'f
spectrometry) flicker factor, dimensionless

"F = detector flicker factor, dimensionless
F = flicker factor for analyze luminescence, dimensionless
C" = flicker factor for analyte luminescence, dimensionless
'L F




-49-


Table IV. Evaluation of Flicker Factors in Emission and Luminescence
Spec trometry


ErITSS ION*
Measurement i F D
Mode e

CW DC DC DC DC

All DC DC DC AC

WI, DC AC AC AC

SM DC AC AC AC


LUM 11ESCErICE*

eas recent LF 'EF 'BF I F iSF DDF
Mode f

CW DC DC DC DC DC DC

AM DC AC AC DC DC AC

UM1 DC DC AC AC AC AC

SM DC DC AC AC AC AC

AM + WM DC AC AC AC AC AC

AM + SM DC AC AC AC AC AC


*The flicker factors are either given by the
a.c. Synchronous Counter Case in Table II.


d.c. integrator case or the





-50-


Table V. Evaluation of Duty Factors in Emission and Luminescence
Spectrometry


EMISSION*
Measurement 1 2
Mode DEl WrlM GD

CL. 1 1 1

AM# 1/2(1)# 1

wrl 1 1/2 1

SM 1 1 1


LUMIN ESCENCE*
measurement 3 D D2
Mlode LM WD.M SB W'.M 0 GD

Cw 1 1 1/2 1 1/2 1

AM# 1/2(1)# 1 1/2 1 1 1

W-M1 1 1/1 1/2 /2 (line)' 1/2 1
1 (cont)'

SM 1 1 1 1 1/2 1

AM + wrJM 1/2 1/2 1/2 1/2 (line) 1/2 1
1 (cont)'

AM + M1 1/2 1 1 1 1 1


Notes

*DEM = 1/2 if the emission is modulated in emission spectrometry
DEI = 1 if the emission is not modulated in emission spectrometry
D1r = 1/2 if wavelength modulation is used and 1 if it is not used
DLM = 1/2 if the source of excitation in luminescence spectrometry
is modulated





-51-


Table V. (continued)


Notes (continued):

DLM = 1 if the source of excitation in luminescence spectrometry is not
modulated
DSB = 1/2 for paired sample-blank measurements
DSB = 1 for sample modulation
DGD = 1 if the detector is "on" during the entire measurement
DGD < 1 if the detector is gated
D = 1 if the exciting source in atomic fluorescence spectrometry is
a continuum source
D.I = 1/2 if the exciting source in luminescence spectrometry is a line
source
DO = 1 if the analyte emission in luminescence spectrometry is
automatically compensated for as in AM
DO = 1/2 if a separate "source off" measurement must be made in
luminescence spectrometry to compensate for analyte emission as
in CW, UM1, and SM cases

Only these two measurement modes are of importance for image device
detectors with image detectors, all duty factors are as shown except
for the case of background emission shot and flicker noise in the All
mode where DEI and DLMl are both as shown in parentheses.

Line means a line interferent; cont means a continuum interferent.




-52-


Table VI. Duty Factors for Pulsed Source-Gated Detector Cases



Pulsed Source-Gated Detectora--No Time Resolution (No Delay Between
Pulsing and Detection)

-t /T.
{t T.[1 e 9 1]1
dAM orEM /f
t [1 e-/fi
9


dGD= tg/tg



Pulsed Source-Gated Detectora--With Time Resolution (Delay of td, s
Between Pulsing and Detection)


-t/T -t /T -t/T
Ti.[l -e ][1 e e
d or d Ep
AM or M -1/fT.
t [1 e 1


dGD = tg/tg



Definition of Terms

t = pulse width of source (assuming rectangular pulse), s

t = gate width of detector (assuming rectangular gate), s

td = delay time between end of excitation and beginning of measure-
ment

f = repetition rate of source (gate), Hz

Ti = lifetime of radiative process, i, s


The duty factors, dAM or dEM, become dGD in the event the radiative
process, i, is not pulsed. These expressions apply to an average;
one must replace tg in the denominator by 1/f for an integrator.











CHAPTER III

MOLECULAR LUMINESCENCE RADIANCE EXPRESSIONS ASSUMING
NARROW BAND EXCITATION



Assumptions


In the derivations to follow, it will be assumed:

(i) that all molecules are in the condensed phase at room tempera-

ture or lower;

(ii) that all molecules are in the zeroth vibrational level of the

ground electronic state prior to excitation;

(iii) that thermal excitation of the upper electronic states is

negligible;

(iv) that the source of excitation is a narrow line, i.e., the

source linewidth is much narrower than the absorption band-

width;

(v) that only one vibrational level in the upper electronic state

is excited:

(vi) that all luminescence transitions originate from the zeroth

vibrational level of the excited electronic state;

(vii) that self-absorption is negligible;

(viii) that prefilter and postfilter effects are negligible;

(ix) that photochemical reactions do not occur;

(x) that only homogeneous broadening occurs.

The expression for the single line excitation rate for induced

absorption used is given by (36)


-53-





-54-


c"' I B ( o)G(vo )dv (III.1)


and the single line de-excitation rate for stimulated emission is given by

E(,o )
f B ('., o)G( uJ )d (I)].2)
c i,L 0 0
-9
where E(,, ) is the integrated source irradiance, Wm 2, c is the
-1
velocity of light, ms B. and B are the Einstein coefficients
-1 3 -1
for absorption and stimulated emission respectively, J m Hz s

ca (.',..) anda (,,v' ) are the normalized spectral profiles of the lower
0' 0 0
-1
and upper levels respectively, Hz- and G(,,,, ) is the normalized

spectral profile of the excitation source. For molecules in the con-

densed phase, free rotation is not possible. The rotational levels have

therefore lost their meaning and the sharp rotational lines of gas phase

spectra merge into regions of continuous absorption. The vibrational

bands may be further broadened by intermolecular forces from the sol-

vent molecules (37). If the only broadening present is assumed to be

homogeneous broadening, then the normalized spectral profiles are

given by


u (I ,'" ) = ('," N)= J ('NJ,v )= /22o (III.3)
o p 0 0 7 _J 0)2 + (Jv/2)2


where ,5 is the absorption bandwidth and ,o is the center frequency. If

the excitation source profile G(,,,v ) is much narrower than the normalized

absorption spectral profile and the source is operating at the line

center, then


a( ) (II .4)
O r J




-55-


The excitation and de-excitation rate therefore become, respectively,

2E
B 2,,E1 ) B .c (111.5)


and

B. ( ) = B (IIl.6)
B I. 2E B E B


3 -1
where c. is the spectral radiant energy density, J n Hz ,and E= E(.. )

For a gas phase molecule, even a laser may not necessarily have a

narrower profile than the absorption profile of individual rotational

lines. For this reason, it will be necessary to convolute the absorption

profile, which is generally best represented by a Voigt profile, with

the spectral profile of the excitation source. Since the source may

also overlap several rotational lines, a summation over all the transi-

tions is required. The absorption rate is then given by

E(>)
^ f BL, 1_u^(,,)G(.,)d. (111.7)
i 1 i

and the de-excitation rate for stimulated emission by an analogous term.

Integrals of this form for a Gaussian laser profile and a Voigt line

profile have been given by Sharp and Goldwasser (36).



Steady State Two Level Molecule

This is a case often valid for condensed phase molecules where

primarily two electronic energy levels are involved in both the radiative

and nonradiative excitation processes. An example would be a highly

fluorescent molecule with little intersystem crossing.





-56-


The energy efficiency for such a process is given by

A20,1i"20,1 i
Y = 1: (111.8)
Y21 A + k
'2j,10 A20,11 + 21


and the quantum efficiency


20,1i
Y (III.9)
Y21
A20,i + k21

where

A20,i = Einstein transition probability for emission (luminescence

transition from the zeroth vibrational level of the

radiatively-excited, 2, electronic state to the ith vibra-
-1
tional level of the lower, 1, electronic state), s-

k2 = nonradiative first order de-excitation rate constant for
-I
same transition given in definition of A20,1i, s

20,1i = frequency of luminescence transition, Hz;

'2j,10 = frequency of excitation transition (absorption transition
from zeroth vibrational level of ground, 1, electronic

state to jth vibrational level of upper, 2, electronic

state), Hz.

The integrated absorption coefficient for the radiative excitation

process, k is given (38) by
2j ,10


I k dv =2j) B n n2 (II1.10)
0 V2j,10 c 1 g2n 1


where





-57-


h-,2j. = energy of absorption transition, J;
-I
c = speed of light, ms ,
3 -1 -2
B10,2 = Einstein coefficient of induced absorption, m J s ;

gk = statistical weight of electronic state, k;
nk = concentration of electronic state, k, m3.

The Einstein coefficients are related to each other (38,39) by

8 3
8nh' n2 1
20 ,li
A20,1i 3 ) B20,1i (111.11)


where
1 -2 -2
B20,1i = Einstein coefficient of induced emission, m3 J- s

n = refractive index of environment (medium), dimensionless.

The Einstein coefficients of induced emission and induced absorption are

related to the electric dipole line strength by


B 2- 2 2, I 1 g (Rel )2 0 )2
20,1i g2) 20,1i % 2 21 I'20(Q) i(Q)
o o (111.12)


2 2
Sil 1 2n 1 el2 0 n g ei 2
B0,2j 2J )S(,2,j --(R2 IgOm10(Q) 1e2j(Q)>I
h2o 9o ( 11.1
o o ( .13)


where

$20'li, S10,2j

o
h
el 2
(R21)

I


2 2
= electric dipole line strength, C m ;

= permittivity of vacuum, 8.854 x 10-12 C2 (Nm2 )-

= Planck's constant, 6.626 x 10-34 J s-
2 2
= pure electronic transition moment, C n ;

= vibrational overlap integral (Franck-Condon factor)

between vibrational levels in two electronic states




-58-


involved in the absorption and luminescence processes

(Q is vibrational coordinate); the Born-Oppenheimer

approximation is assumed to apply here;

e(Q) = vibrational wave function which is a parametric

function in Q, the nuclear coordinate, dimensionless.

The concentration ratio of state 2 to state 1, n2/n1, is given by


2E(v O2)
n B 1(,2j
102 c6v ( (111.14)
n1 A + B 2E(v10,2j
1 A 20,1i + B2j,10 102 +k
i C nc5,v k21


for steady state conditions and for the condition of negligible thermal

excitation (kl2 :t 0). In Eq. 111.14, E(v10,2j) is the source irradiance

(integrated spectral irradiance) of the exciting line and 5v is the

half-width of the absorption band undergoing the transition, e.g., for

a gaseous molecule, as OH (12); the absorption bands will be of the order
-l
of 0.1 cm whereas for a molecule in the liquid state, all rotational

and often even most of the vibrational structure of the electronic band

is lost resulting in a broad band, such as 6v > 10 nm. Equation 111.14

can be rewritten in terms of the quantum efficiency (see Eq. 11.9).


n pB 2E(v10,2j
2 10,2j nc6.,,
(Ill.15)
n1 A201

Y + B 2j,10 E ,2j
21 nc6v

By utilizing the definitions of the A's and B's (see Eqs. 111.11-

111.13)





-59-


1~~,,a


A = =A
20,ii 2j,10
l


1li (Q 2 3 i
I i 20,1i


(111.16)


2 3
2j(Q) l1 lo (Q): 2 j, 3
2i 10'2i 10


where A2j,10, the electronic-vibrational transition probability at the

absorption frequency, is


3 3
A2j, 1 _16n_ I el2 23 2
2je c g2


(111.17)


where all terms have been previously defined. If we now use the follow-

ing substitutions for simplicity


V20,ii =le20(Q) leli(Q) 2 = eli(Q)Ie20(Q)- 2


V2j,io I-e2j(Q)Io10(Q) 12 I. 10(Q)162j(Q)1 2


A21 A2j,10


B21 = B2j,10


812 = B10,2j


E = E(10,2j)


then


cA2 20,li"20,1i
3
2B V E Y
+ 21 2j,10E Y21 2j,10
2120.
1


(III.18)


n2
n





-60-


Simplifying Eq. 111.18 by use of the relationships between B12 and B21

(B21g2 = B21g1) and dividing numerator and denominator by V2j,10 gives


21 2 12j,10
ncA G. | 3 )
gn L2 .
-- 2E123 (11.19)
92n1 1 + [2EYcA B i 2j,3 .
1 + 2 21JI -2j,10
2j,10 21 6l V 1
S20,11 20,1i

According to Strickler and Berg (39),

T 3
S20,1i "20li f F(v)dv -3 -I
S= <- >- 1 (111.20)
f \ -3 L AV
S 20,li F(V), d
-3 -1

where F(,) is the luminescence profile function and <.-3 >-A is the
L AV
-3
reciprocal of the average value of .L in the luminescence spectrum.

Because V20,i = 1, i.e., orthonormal complete set, Eq. III.19 can be

rewritten as

2EY21B21 3 -3
91n2 cA216" v 2j,10 "L >AV
22 FY B -(III.21)
92n1 1 I2EY21B21 3 -3
V2 + 1c 2j,10<'-L 'AV
V2j,10 ncA21 v 2j,10 L AV

If as in atomic fluorescence (4), E*, a modified saturation spectral

irradiance, i.e., E* is related to Es, is defined as

cA21
E* B21 (III.22a)
12 B21Y21

and if c2j,10 is defined as


J 3 -3 (I22
2j,10 "2j,10




-61-


then

2E
1 2_ (111.23)
92n1 E 12 2E
+
J2j,O '10 '

The fluorescence radiance expression (4,40) for a two level system

is given by

2E h" 2j, 10 g1n2
BF = )Y ( ,--,)nl Jc 0B 1 )B (111.24)
P 4 21 1 c 12 g2n1


where I is the fluorescence path length in the direction of the detector.

Substituting into Eq. II.24 for the ratio g1n2/g2n1 from Eq. III.23 and

for n1 in terms of n2 from Eq. 111.23 gives

,h B12qlnn2
B = (- )Y E* ( .)( 2' 2 ) (111.25)
F 4n p21 '12 c 92"2j,10


By evaluation of Y A20,1i (combining Eqs. 111.16 and 111.20)
1
A
TA 2j,10 (11.26)
L 20,li
i `2j,10

and by substituting for Y in terms of Y1 (Eq. III.8) and for E*
P21 (
(Eq. III.22a) into Eq. III.25, BF becomes


BF = (-)2 A20,11h 20,1i (11.27)


which is the expected expression based upon previous derivations for

atomic fluorescence (4,41). However, it is interesting to stress that

BF is independent of the vibrational overlap integrals.

Evaluating n2 in terms of nT, where nT = n1 + n2 total concen-

tration of molecules in all electronic states gives





-62-


B = () A hv ( 2E nT 1 (I.28)
F 4n 20,1i 20,i g1 2E g E* (111.28)
S(1 +2)( ( +
92 9g2 42j,10

which has exactly the same form as the 2-level atom fluorescence radiance

(1-4).


Steady State Three Level Molecule

Molecules in the condensed phase (solids mainly) as well as some

molecules in the gas phase (depending upon pressure) must be treated as

at least a 3-level system, e.g., a ground singlet, 1, a first excited

singlet, 3, and a first excited triplet, 2. The same approach as in the

previous section will be carried out.

Assuming the upper level, 3 (1st excited singlet) is being radia-

tively excited and assuming the nonradiational excitation rate constants,

kl2 and kl3 and the radiational rate constant A32 are negligibly small

(here only the electronic states are listed in the subscript, not the

vibrational levels), then the ratio of concentrations for state 3 to

state 1, n3/n,1 is


nj (y A + k23 + k )
nR c62 20,1 i 23 21
J _I
1 L 3j,10 + A k + k 31 A + k+k A-1 k23k32
L nc, 30,1i 31 32 20,1i k23 +k21 2332

(111.29)

where all terms have the same definitions given previously except the

levels involved may differ and E = E(3j,10).

The definition of the power, Y and quantum, Y, efficiencies for

electronic transitions 3 1 according to Lipsett (42) and Forster (43) are





-63-


A
S30,li"30,i
Y = (111.30)
31 [A +k +k 23 32
30,11 31 32 + 2 ex
A + P + k
20,1i 21 23




30,1i
Y k (111.31)
31 kk
kA +k2 +k
30,1i 31 32
31 3 3 20,11 k21 + 23



where e, is the excitation frequency with appropriate subscripts. For

the 2 1 transition excited by radiationless transitions from level 3,

the power and quantum efficiencies are given by


Y P .=32y (111.32)
P21 32 P21


Y21 = 32Y21 (111.33)


where ,32 is the crossing fraction (also termed quantum yield of inter-

system crossing or triplet yield) and is given by


32
-32 k k (111.34)
A + k + k2332
30,li 31 32
SA20li + k21 + k23


where y 2is the radiative power efficiency and y21 the radiative

efficiency, given respectively by


rA
S20,i 20,1i
yP A2 (II.35)
21 T[A + k 20, 1 kj





-64-


and

A20,1i
y = i (III.36)
21 A + k + k
S20,1i 21 23

for 2 1 luminescence excited indirectly. Combining Eqs. III.29 and

111.31 gives


293B3j,10 E Y31

gc, C A' A
gn 91 QC,, 30,1i
--g--. Y---A--l ( I I I 3 7 )
1 2B 10 E 31 11 .37)
+ 1
30,li

Substituting for L A from Eq. 111.16 (replace 2 by 3 in all terms)
30,li
and making substitutions of B13 = B 103j and B31 = B ,, and A =

A3j,10


2g3B31Y31EV3j,103j,10

n3 lnC A31 .V 30,li'30,li
n + 2B (31 EV 3i 1 (111.38)

ncA31 6.L v30,li 30,1i)
1

Using the Strickler and Berg (39) approach (see Eq. 111.20) and the

definition of E* and i as
-1 3 3j,10

cA3
E* (III.39a)
'1 3 B31 Y31


3 '3A (13.39b)
3j,10 -3j,10L AV3j,10.39b)





-65-


then n3/n1 is given by


.93) 2E
n3 g91 ,i.
n E*
2E 1 3
'-3j,10


(111.40)


The fluorescence radiance for the 3 1 fluorescence transition is

given (4,40) by


B = ( Q)Y (2E-)n 1( v 3)B P[1
F 4n p31 1 c 10,3j,


gln3
93n1


(111.41)


Substituting for n1 in terms of n3 and for n3/n1 from Eq. III.40 gives


B F -= (-L-)n A h, 0
F 4 3 30, 1 i 30, 1 i
]


(111.42)


which is identical in form to the expression for the 2-level case (Eq.

111.27). Substituting for n3 in terms of nT (nT '- n1 + n2 + n3) can be


done using Eq.


111.40 for n3/n1 and Eq. 111.32 below for n2/n1


n 1
2 -
n1


(I11 .43)


and so


n3


E*
2E '13
1+ '3.j,10 +
1 + +
g 2E,
-) (-)
I j 1,%.


k32
SA20,1i k21


(111.44)


and


Sk23





-66-


B = ( i) A h -
F 4= 30,1hi 30,1i
3-1
1-3
f n ]

SE* T (111.45)
2E '13
1 + nv 3j ,I0 k32
+g3 A + k + [
S(- ) 20,li 21 23


where the subscript on BF indicates the emission process (above) and the

absorption process (below).
The radiance for the 2 1 phosphorescence transition (assuming)

conventional 1 -3 excitation) is given (4,40) by


B = )Y ( hu0,3j g[1 ln3 2E
21 4n 21 c 1)0,3jnl gn 1
1-3

where Y is the quantum efficiency for luminescence from level 2 while
P21
exciting level 3. Substituting YP21 (Eq. 111.32) and n1 and n3/n1

(Eq. 111.40) gives the expected relationship for 1 3 excitation


B = (-L)n A hu (111.47)
P 4n2 20,li 20,li
2 -1 1
1 -3

and substituting for n2 in terms of nT gives

B = (0 ) X A hv
Bp- 4n A20,ih 20,1i
1-3


(111.48)





-67-


The final case of potential interest is radiative excitation of

state 2 directly from state 1. In this case, Bp is given by


B = ( y ( l0, .n [
BP2 1 4n 21 c )B10,2j 1

1-2


91n2 2E
g2n1 ,-.,
- E^K-)
92g n TI 'X .'


where E = E(..10,2j). The ratios n2/n1 and n3/n1

(2 becomes 3 and 3 becomes 2 in Eq. 111.40) are


(III.49)


for this excitation case


(2 2E

E E*
2E 12
" ,2j 10


n3
"3
n1


k23

30,1i k31


(III.50)


(II1.51)


k32]"1
'32


Substituting for n2/n1 (Eq. III.50), for n1 (Eq. III.51) and for Y

(Eq. III.32) gives


B = ()n
P 4n 2
2-1
1-2


A20,li 20,1i


and substituting for n2 in terms of nT (nT ~ + n + n3 using Eqs.

III.50 and [11.51)


and


(111.52)




-68-


Bp (4n) A20,i i20,1i
2-1 1
1-2

I "nT n
E* 1 (III.53)
2E + 12
1 "+ 2j ,0 + 23

91 +T


A rather trivial case involves excitation of state 2 from state 1,

intersystem crossing 2 to 3, and fluorescence from 3 to 1. This case is

a form of fluorescence. The radiance expression for BF is


B = (-) A 3 h
F 4 30,1130,i
1-3
nT
E* (III.54)
2E + "12
A + k + k 2E + ?
30,1i 31 32 1+ .,v 1
12] 11 +2.j ,10
1 + 11 +
k 2E
k23 1 2 2E


A nontrivial but analytically unimportant case is E-type delayed

fluorescence, DF, where excitation of 3 from 1 occurs followed by inter-

system crossing 3 to 2, reverse intersystem crossing 2 to 3, and, finally,

delayed fluorescence from 3 to 1. The quantum efficiency and power

efficiency for this process is


Y 31 32 23Y 31 (111.55)
p31 '323p31


where 32 is given by Eq. 111.34, and <23 and y3 are given (42) by
233

=32 k 3 (111.56)
2 A2 i + k2 + k2
20,li 21 23





-69-


and


(111.57)


S30,1 i '30,1i

A301i + 31 k3213j,10
1


Substituting for Y


into Eq. II1.41 and for nI and n3/n1 as previously


for the case of 1 -- 3 excitation and 3 -- 1 fluorescence, gives


:k23k32

[A30,li +k31 k32][A20,li +k21 '23]
i 1


g3
91
1l


g3 2E
g1 n.T

+ 3 2 ,32 2 2E
91 A +k +k n.1i 1
g A20,1i +21 k23
l r


30,1i '30,1i


(111.58)


Em
+ '13
'3j .10


where E = E(..0,3j )



Limiting Cases of Steady State Excitation


In all cases given, high implies that E(..) -.> E*6..n/2Y and low

implies that E(.,) << E*6..n/2.. Limiting expressions are given for cases

of analytical utility.

For a two level molecule, if the source irradiance is low, then

BF (see Eq. 111.28) becomes


B (Io) = (F)T
2-1 '
1-2


A hi 2 n .)10,nIJ )2j 10)
20,1i h '20,1i g1 ) T 2 E* j
'12


and if the irradiance is high, then BF (see Eq. 111.28) becomes


BF (hi = (-)T A ,ih 2 0,1i)( "
212 1 g1
-. +
1 2 g2


YP31


(II .59)


(111.60)





-70-


For a three level molecule assuming 1 3 excitation and 3 -, 1

fluorescence, if the source irradiance is low, then BF (see Eq. 111.45)

becomes


BF(lo) = () A h30lih3)n 2E *10,3j '3j,10 (111.61)
F 4P 30.li 30,li TI E* 11v
3-F 1 1 3
1-3

and if the source irradiance is high (see Eq. III.45), BF becomes


B (hi) = (- ? A h nT ( 1.62)
F = 30, 3,i g k (111.62)
3-1 1 1 32
1-3 g 3 A +k +k
20,11 21 23

For a three level molecule, assuming 1 3 excitation and 2 1

phosphorescence, if the source irradiance is low, then Bp (see Eq.

I11.48) becomes


Bp(lo) = ( )A h(-3) n k32 HE(Ul0,3j 3j,10
2-1 1 )A +k +k l u
1-3 20,11 21 23 13
31 (III.63)

and if the source irradiance is high (see Eq. III.48),the Bp becomes


B (hi) = (h) n Tk 32 d 1
P 4 201n20.,132 g 1
P 2-1 4ni "A +k +k 1 32
1-3 i 20,li 21 23 1 +_+
93 TA +k +k
3 A20,1i 21 23

(III.64)

For a three level molecule assuming 1 2 excitation and 2 1

phosphorescence, if the source irradiance is low, then Bp (see Eq.

III.53) becomes





-71-


B (lo) = (
P2-1 4
1-2


A .h f 2E (v102 2j, l10
20,11i 20,li lg nT E* n ..
S"..12


and if the source irradiance is high, then Bp (see Eq. 11.53) becomes


S20 ,11h20,1i
1 1


+ 23
7A +k +k
. 30,1i +k31 32
1


Steady State Saturation Irradiance


The saturation irradiance is that source irradiance resulting in a

luminescence radiance equal to 50'. of the maximum possible value. For

a 2-level molecule, it is given by


Es(Oj)
10,2j


E* n-i .,
12
2j33lO 'L -AvV2
2 2j,10' "L 'AV 2j,10


I 1


(11 .67)


For a 3-level molecule (1 3 excitation), it is given by


E*
1 g 13 ) n2 ,
v3 1 V j 10 2 3 j ,-3 -
-j 2 3j 10' L >AV


Es('10,3j)


91
1 + +
93


(111.68)


k32

A + k + k
20,1i 21 23
1


For a 3-level molecule (1 2 excitation), it is given by


E (10,2j)


gl 12 2 -3
2 V 2j,10. 2 1 "L 3AV


(111.69)


S30,li + 31 + k32
1


(111.65)


B (hi) (-)
2-1
1-2


(111.66)


91
1 + +




-72-


However, Eq. 111.69 can be simplified further since the final term in

the denominator will generally be negligible and so reverts to the 2

level expression in Eq. 111.67.

For a typical organic molecule at 2980C, E* 1.8 x 10-6 W cm-2Hz-1
6 -2 "12
(6 106 W cm2 nri) (assuming Y21 = 1 and k21 = 300 nm) or E* .
-5 -2 -1 7 2 1 12
1.8 : 10 W cm Hz (6 x 10 W cm nm ) (assuming Y21 = 0.1 and
3 -3 V,
'21 = 300 nm). Assuming 22j l0 S10 10 Hz (gaseous molecule) or .v 101 Hz (molecule in liquid

solution), then E s(;10,2j) 10 kW/cm2 for the gaseous molecule and

Es (.1O,2j) 105 kW/cm2 for the molecule in the liquid state assuming

Y2 = 1 and 121 = 300 nm. For a 3-level molecule, Es (,10,3j) will be

smaller than ES( ,2j) by a factor k /T A0 + k1 + k which will
10,2j 32 20,li 21 23
5 7 1
be -.10 -10 for most molecules (44,45).



Nonsteadv State Two Level Molecule

If the duration of an excitation source pulse is comparable to or

shorter than the excited state lifetime, then the steady state approach

does not hold. The nonsteady state treatment of two level atoms has been

given by de Olivares (5). It is only necessary to slightly modify the

expressions she has given for atoms, so no detailed solution will be

given.

From Eq. 1I.28,it is possible to define a steady state concentra-

tion of n2, n2ss. This is given by

n
"T
n2ss E* (111.70)
g1 g "12
(1 + -)cp102j +
2 10,2j 2 -2j,10





-73-


where the spectral radiant energy density, ), has been used. For a

rectangular excitation pulse, )(t) = co for 0 < t < t and o(t) = 0

for t > to where to is the pulse width, s. The concentration of n2 as

a function of time, n2(t), for 0 s t to is


n2(t) = n2ss[1 exp(-(a + bo )t)] (Ill.71)

where

A21
a = + k21 (111.72)
2j,10 '2

and


b = B12 + B21 (111.73)


For low irradiance cases, the growth of n2 population is controlled by

the luminescence lifetime, a-1 As the irradiance exceeds the saturation

irradiance, the growth of n2 population is more rapid. If the pulse

width is long compared to the lifetime, the steady state concentration

of n2 is reached.



Nonsteady State Three Level Molecule


The solutions for a three level atom under nonsteady state con-

ditions have been given assuming thermal equilibrium between the two

upper levels (5). This situation will not apply to molecules, as the

relative populations to the two upper levels is also dependent on the

intersystem crossing rate constant. Collisions are not required for

population of the triplet from the singlet. Starting from the rate

equations assuming excitation of level 3 from level 1,







dn A31
dt -(B31 (t) + 31 + k3 + k)n3 +
43j,10 31 32

(B13"13(t) + k12)n1 + k23n2 (111.74)

and
dn2
dt ( A li + k21 + k2)n2 + k32n3+ k12 ( .75)

It will be assumed that thermal population of levels 2 and 3 is negligible

at room temperature or lower, making kl2 = kl3 = 0. It will also be
assumed that intersystem crossing from level 2 to level 3 is negligible,
making k23 = 0. The following terms are defined to simplify Eqs. 111.74
and III.75.
A31
a 3 + k3 + k2 (111.76)
3 j,10 k31 32


a2 Y A20, + k2 (111.77)


b3 = B31P13(t) + B13p13(t) (III.78)


B = B13"13(t) (III.79)

Using D to denote the differential operator, Eqs. III.74 and 111.75 may
be written as 111.80 and 111.81, respectively, after substituting
n1 = nT n2 n"3

(D + 3 + a3)n3 + Bn2 = BnT (III.80)

D + a2
D+a2
-n3 + n2 = 0 (111.81)
32





-75-


Eliminating the n3 term from Eq. 111.80 by multiplying (D + b3 + a3)

times Eq. 111.81 and adding the result to Eq. II.80 gives

(D + b3+a3)(D + a2)n2 + Bk32n2 = Bk32 (111.82)

The solution to the homogeneous differential equation of the form

of Eq. 111.82 for 13 = Po for 0 t t to is


n2(t) = Clexp(-a2t) + C2exp(-.3t) + C (111.83)

where

X X2 4Y
"2 2 (111.84)


S + 2 4(111.85)


and
X = b3 + a3 + a2 (111.86)


Y = (b3 + a3)a2 + Bk32 (111.87)

The particular solution of the nonhomogeneous equation gives C as
0
k n
C = ( ) (Ill.88)
o a 2 k32 a + b1.88)
"2 k32 3 3
a2 B

Using the solution for n2(t), the solution for n3(t) may be found

using Eq. 111.31. The arbitrary constants C1 and C2 are evaluated from

the boundary conditions n2(0) = 0 and n3(0) = 0. This gives the final

expressions for n2(t) and n3(t) as


n2(t)= n 3 exp(-.:2t) + 2 exp(-3t) + (111.89)
x2(t) 2ss -
X2_4Y ,X2-4




-76-


3(2- 2 2(a2-a 3)
n3(t)= n3s exp(-2 t) + (a2-3) exp(-a.3t) + 1 (III.90)
La 2 -4Y a2 -4Y


wher2 n3ss is given by Eq. III.44 and n2ss is given by


n k32n3ss (111.91)
2ss V~A + 1-
A20,1i :21
i


At low source irradiance, a2 % a2 and a3 a3, where a2 is the reciprocal

of the level 2 lifetime (phosphorescence) and a3 is the reciprocal of

the level 3 lifetime (fluorescence) which is the conventional low ir-

radiance case (40).

In order to better understand the expressions for n2(t) and n3(t),

calculations using literature values (44-46) for transition probabilities

and rate constants were performed and plotted for three limiting cases.

Benzophenone represents the case of a molecule with a poor fluorescence

quantum efficiency (.010-4) and a large phosphorescence quantum efficiency

(-0.9). Fluorene represents the case of a molecule with a moderate

fluorescence quantum efficiency (.'.0.45) and a moderate phosphorescence

quantum efficiency (-10.36). Rhodamine 6G represents the case of a high

fluorescence quantum efficiency (-1) and a small phosphorescence quantum

efficiency (-10 3). Results of calculations of log(n2/nT) and

log(n3/nT) versus log(t) are plotted for benzophenone, fluorene, and

rhodamine 6G and shown in Figures 4, 5, and 6, respectively. In all

cases, the value of n2/nT approaches the steady state value of n2/nT

more slowly after n3/nT reaches its steady state value. As the source

irradiance increases above the steady state saturation irradiance, the

time required to attain steady state decreases. If the source irradiance





-77-


is less than or equal to the steady state saturation irradiance, the

value of n3/nT increases until it reaches a value predicted by the 3-

level steady state model. If the source irradiance exceeds the steady

state saturation irradiance, the value of n3/nT will also exceed the

3-level steady state saturation value of n3/nT until n2/nT saturates.

Until the concentration of level 2 approaches steady state, levels

1 and 3 are acting in a fashion similar to the 2-level model. The

2-level model predicts a saturation irradiance approximately 105 times

higher than the 3-level model for rhodamine 6G, and it is observed in

Figure 6 that at 106 Es, the concentration of level 3 is close to

saturation. For benzophenone and fluorene, the 2-level saturation

irradiance is greater than 107 times the 3-level saturation irradiance,

so no saturation of level 3 is observed. It should also be noted

that for the pulse widths of available lasers (-l1 ps for flashlamp

pumped dye lasers and -10 ns for nitrogen laser systems), it is not

possible to saturate level 2 (triplet) of most molecules in a single

pulse without focusing to a very small area. For lifetimes longer

than the time between pulses, the effect of short pulse width is

partially offset because the triplet population does not decay to

zero between pulses. This will decrease the required irradiance by

approximately the factor 1 exp(-l/fT ), where f is the sourse

repetition rate and T is the triplet lifetime (see Table VI).

Returning to the terms in Eqs. 111.89 and III.90, the coeffi-

cients of the exponential terms may be discussed. The factor -r 3/ X2-4Y

in Eq. 111.89 is approximately -1 and the factor r2/'X2 4Y is

approximately the ratio of the fluorescence rise time to the phos-

phorescence rise time. As the source irradiance increases above the
























V-





S-
Cj




0
C0


O LO
J S3-
w > -

0 Cn
C E
-, i 0






4-J
S.S.- L




o eu
CL L-




> O --**


0 ,-




a) C"J
E


n E

> M I --
- "- 0 -Z




0 'LA a C) C C) CC X 00
M0 0 -- - C-
S ,-- iin, C0 -- .,



,O II 00 m--


O S- 4- X X X X X C D
o f*
- U *
> C-C\J --0 ( -
tD ii m * * < l OO M
-R U e --- -- r-- sa II L :
aC w <- Om LD
C a II II II II II in .-.
>U U c II O

0 0

0. t
E -- 0 --
a .m OinC
r-









LL
CQUD 1 1 1 1 1 i--11 i






-79-


o:
*
I


o* *
**a


* *
o* *












*
0











0
0.




















\ \ *
\j C



LL

















o 0 Q


I l I I I






















uIl








C



OS
C

4-
s- CD

00


> Q)


E
0l) iA' 0


L- I-

0 a




4- 4- C\J


-, c-
C C r'
'r0

E



i- '0 *. *- N ,
- IA .'.. ,- 2 X\J

0.-- o i-n i UI v- .

0) cLD I"C "I I -- E
O i OOOO --
0 0.- .- i-- -- -- X C
t-- C) ::3 o
L -) x x x
- ro E x
> C -* Nj L CCo C\j --
ra *- 3 * * o r
-C U L C\J :T C'. O II c2
o0 Lu Col) 00
co ro 11 11 11 1I II -C\J I L
4- u 0 I1 0
.- o -- j -- *--I1 'o0 4-
a0 C mr)mre)C\Jm >i i
o 1l **c* -K -z -0 -, *z -j 'C Lu U"
0 O<
0. = 0
E E-
aj ro Al M






s-
:3
cn
LI.. n




-81-


Ie f

\ . 7

\ .
\ N
S * "

S* *. S C




S..


. S
N\ \. KO I



.* \\\,


D ,- C-

.. ( Lri C c v T'
* -e N ^ ) 0 -r- ^ N / ) 9 1






















Ln






C)
3

r0









C -
O 0
ic U




cu 4-





















E
4-


OC







r- 0







0 C C.-










0 *r-- X in (U "C .- .
a-- C-
" 4-







- -I I..
o \ aa)
o oI-
















4. l r- c-
4-- U L 0 LO I I
o =3 . c.


-0 U

wn



S-
CM L

s' 2 c-
.- .. "*"








1- C*r- 0





a) ULi i- 01- i-
Oi jill lII I -II L i/ 5-
4-u ni C) I 0







1- => 3C








LL-
a) n Od1 4






-83-


(~Q


N(I/a N) DO-,' (-Nle'/N)0-)Oi


0e
eS




0 *
0.


*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
^ *
*
*
*
*


,\
*


0




O















I


' LU

















co











O





-84-


saturation irradiance, phosphorescence rise time decreases. When the

n3 concentration reaches steady state, the rise time ratio term con-

tribution approaches zero as it is multiplied by exp(-a3t). The terms

L3(a2 a2)/(a2X2 4Y) and (2(a2 c3)/(a2X2 4Y) in Eq. II1.90 are
close to the same value and opposite in sign; this value is the ratio

of the excitation irradiance to the steady state saturation irradiance,

E/Es. As time increases, the term exp(-a3t) decreases the absolute

magnitude of the negative term and the concentration of n3 increases

to the value allowed by the positive coefficient of exp(-n2t). As

exp(-a2t) decreases (time approaching the lifetime of level 2), the

value decreases, and the steady state concentration of level 3 is

reached.

Thus far, only the relative populations of the levels have been

discussed. The expression for the luminescence radiance may be obtained

by substituting Eq. I11.89 for n2 in Eq. III.47 and substituting Eq.

III.90 for n3 in Eq. III.43.



Conclusions


The major conclusions which can be made from the previous expressions

are

(i) the radiance expressions for molecular fluorescence are similar

to those for atomic fluorescence (2-4), and reduce to the case

of atoms if tne term c is equal to unity;

(ii) for low source irradiances, the luminescence radiance depends

directly upon the source irradiance and the quantum efficiency;





-85-


(iii) for high source irradiances, the luminescence radiance is

independent of the source irradiance and the quantum effi-

ciency;

(iv) for all cases, the fluorescence radiance depends directly upon

the total concentration of analyte, nT;

(v) for all cases, the fluorescence radiance depends directly upon

the transition (emission) probability for the measured process;

(vi) for the 2-level case under saturation conditions, the total

concentration, nT, can be determined by absolute measurement

of the steady state BF-value, by knowledge of A20,1i' g91 2'

and by measurement of the cell path length in the direction of

the detector;
3 -3
(vii) the product v. -L "AV term, occurring implicitly in the factor

in all radiance expressions will be not greatly different from

unity;

(viii) the V-term occurring implicitly in the i-factor in all general

radiance expressions, accounts for the overlap of vibrational

levels during the excitation transitions as well as for the

fractional portion of the electronic absorption band being

excited, e.g., with a gaseous molecule, one could excite only

one of the vibrational levels of the excited electronic state

and so only a fraction of the absorption band is excited

(actually this factor could be separated out of V and designated

Jf/f where f is the oscillator strength of electronic transi-

tion and ,f is the oscillator strength portion attributed to

the excitation transition);





-86-


(ix) the saturation irradiance, Es, for a 3-level molecule at room

temperature is 105 to 107 less than for a 2-level atom or

molecule at any temperature or for a 3-level atom or molecule

at high, e.g., flame, temperatures; because of the greater

half-widths of molecules, saturation can be achieved either by

a high spectral irradiance over a narrow line width or a low

spectral irradiance over the broad absorption line width

assuming the same effective irradiance (within the absorption

band) reaches the molecule of interest, i.e., for narrow source

line excitation, E of the laser source must exceed 2Es/'aser

and for broad band excitation solutions, the requirement

for saturation is that E of the laser source must exceed Es

the saturation spectral irradiance equal to 2ES/navabs;

(x) assuming saturation is reached, direct excitation of the trip-

let state is nearly as efficient as conventional excitation of

the first excited singlet state with intersystem crossing to

the first triplet state; therefore, visible cw Ar ion dye

lasers, assuming they can be focused down to 10 u m to achieve

.IMW/cm2, can be used to excite many molecules with no need for

doubling; if 'IMW/cm can not be achieved and if the phosphor-

escence quantum efficiency is considerably less than unity,

then saturation of the triplet level (essentially a 2-level

case) by direct excitation is not possible;

(xi) if the source irradiance exceeds the saturation irradiance, the

steady state condition is reached in a shorter time;

(xii) the steady state concentration of n3 (singlet) may be exceeded

under pulsed excitation conditions. The optimum measurement





-87-


system for fluorescence is a pulsed laser where the high peak

power may be utilized to increase the fluorescence signal;

(xiii) due to the relatively long time required to reach steady state

in level 2 (triplet), saturation of the triplet level using

pulsed lasers will not be possible without focusing the laser

to small areas to increase the irradiance to a level of

(5 /t )ES where p is the phosphorescence lifetime, to is the

pulse width, and Es is the saturation irradiance; this term is

obtained from 1 exp(t /T )= t /T for t /Tp I and the

factor of 5 from the fact approximately five lifetimes (rise-

times) are required to reach steady state.











CHAPTER IV

PULSED LASER TIME RESOLVED PHOSPHORIMETRY



Introduction


Time resolved phosphorimetry was first demonstrated as a means of

chemical analysis by Keirs et al. (47). They resolved a mixture of

acetophenone (T = 0.008 s) and benzophenone (Tp = 0.006 s) at concen-

trations in the range of 10-3 to 10-6 M. O'Haver and Winefordner (48)

discussed the influence of phosphoroscope design on detected phospho-

rescence signals. St. John and tinefordner (49) used a rotating can

phosphoroscope system to determine simultaneously two component mixtures.

O'Haver and Winefordner (50) later extended the phosphoroscope equations

to apply to pulsed light sources and pulsed photomultiplier tubes. The

expression for the duty factor (50) applies to a d.c. measurement

system. The expression for the duty factor using a gated detector

(boxcar integrator) is given in Table VI.

Winefordner (51) has suggested that the independent variability of

gate time, t delay time, td, and repetition rate, f, of a pulsed

source-gated detector along with the spectral shift toward the ultra-

violet (52) when using pulsed xenon flashlamps should make such a system

optimal for phosphorescence spectrometry.

Fisher and Winefordner (53) constructed a pulsed source time re-

solved phosphorimeter and demonstrated the analysis of mixtures via time

resolution. This system was modified to use a higher power xenon


-88-





-89-


flashlamp with which O'Donnell et al. (54) time resolved mixtures of

halogenated biphenyls and Harbaugh et al. (55) measured phosphorescence

lifetimes and quantitated drug mixtures (56). Strambini and Galley (57)

have described a similar instrument for phosphorescence lifetime

measurements.

The emphasis in pulsed source time resolved phosphorimetry has been

on selectivity rather than sensitivity or precision. Johnson, Plankey,

and Winefordner (58) compared pulsed versus continuous wave xenon lamps

in atomic fluorescence flame spectrometry and found the continuous wave

xenon lamp to give 10-fold better detection limits. The pulsed xenon

lamp had been predicted to give better detection limits (15). The con-

tinuous wave source had an 85-fold larger solid angle. The linear

flashlamp used was 2 in long, making it difficult to transfer the

radiant flux to a small area. This is a critical problem in phosphori-

metry because the sample height is less than 1 cm. Johnson et al. (59)

attempted to overcome this problem by pulsing a 300 W Eimac lamp (Eimac,

Division of Varian, San Carlos, Calif. 94070). The improvement in S/N

failed to materialize due to instability of the pulsed lamp and due to

the high d.c. current required to maintain the discharge between pulses,

which reduced the fluorescence modulation depth. In phosphorimetry, such

a source would give extremely high stray light levels caused by the

cylindrical sample cells. A point source flashlamp is now available

(Model 722, Xenon Corp., Medford, Mass. 02155) and would appear to

offer the best compromise as a pulsed continuum source for phosphorimetry.

The point source should allow an increase in the useable radiant flux

transferred to the sample.





-90-


A second major consideration to signal levels when using pulsed

sources is the pulse repetition rate, f; at constant peak power, f

controls the average power of the lamp. Previous investigators (54-56,60)

have operated xenon flashlamps at a maximum f of 0.2 Hz. From the

equations in Table VI,it can be seen that the term, [1 exp(-l/fr )],

in the denominator decreases as fp becomes greater than unity. If all

else is constant and T = 1 s, the signal level is 20-fold higher at

20 Hz than at 0.2 Hz. This is the major reason for low signal levels

observed with pulsed source phosphorimetry when compared to conventional

phosphorimetry.

One of the fundamental limitations with continuum sources, whether

continuous wave or pulsed, is that only a small fraction of spectral

output is useful for excitation of phosphorescence. Even assuming fast

collection optics and wide-band interference filters, the useful radiant

flux transferred to the sample is still only a small fraction of the

total spectral output. Using higher power sources is difficult due to

stray light problems. The ideal case would be a source of high intensity,

tunable, monochromatic radiation. Such a source is the tunable dye

laser.

The dye laser is the finest available excitation source for both

atomic and molecular luminescence spectrometry due to its high spectral

irradiance, small beam diameter and divergence, and wavelength tunability.

The theory of laser operation is given in many texts (61-63). Allkins

(64) and Steinfeld (65) have reviewed many uses of lasers in analytical

spectrometry. Both continuous wave (66) and pulsed (67) dye lasers have

been utilized to obtain excellent detection limits in atomic fluorescence

flame spectrometry. Dye lasers have been applied to molecular





-91-


fluorescence spectrometry (68-70), photoacoustic spectrometry (71),

Raman spectrometry (72), and Coherent anti-Stokes Raman spectrometry

(73). Fixed frequency lasers such as the nitrogen laser (74), the

He-Cd laser (75), and the argon ion laser (76) have also been utilized

in molecular fluorescence spectrometry.

Although dye lasers have been used extensively in studying elec-

tronic and vibrational parameters of the triplet state (77-79), no

analytical applications of dye lasers in phosphorescence spectrometry

have been reported. Wilson and Miller (80) used a nitrogen laser to

time resolve the spectra of a mixture of benzophenone and anthrone, but

reported no analytical figures of merit. This work reports analytical

figures of merit for laser excited time resolved phosphorimetry of druns

and compares the use of two different lasers (pulsed nitrogen laser and

flashlamp pumped dye laser) as excitation sources.



External Heavy Atom Effect


Analytical Applications


The first suggestion of the analytical utility of the external

heavy atom effect was from McGlynn et al. (81). Hood and Winefordner

(32) and Zander (83) found improved detection limits for several aromatic

hydrocarbons using glasses of ethanol and ethyl iodide. The use of

quartz capillary sample cells with snows of ethanol or methanol water

mixtures permitted the use of large concentrations of halide salts in

the solvent matrix (84). Lukasiewicz et al. (16,17) reported improved

detection limits in 10% w/w sodium iodide solutions. Other investi-

gators (85,86) have reported on the analytical utility of sodium iodide





-92-


in 10/90 v/v methanol/water at 77 K and at room temperature (87-89) on

filter paper.

Rahn and Landry (90) found a 20-fold enhancement in the phospho-

rescenceof DNA when silver ion was added and attributed the effect to

silver ion acting as an internally bound heavy atom perturber. Boutilier

et al. (18) studied the effect of silver and iodide ions on the phos-

phorescence of nucleosides and found silver ion to improve detection

limits 20 to 50-fold. Other metal ions (Cd(II), Hg(ll), Zn(II), and

Cu(II)) have been studied as heavy atom perturbers (91-92) at 77 K and

Ag(I) and T1(I) at room temperature on filter paper (19,93-94).



Theory


The external heavy atom effect was first observed in 1952 by

Kasha (95) when the mixing of l-chloronapthalene and ethyl iodide, both

colorless liquids, gave a yellow solution. The color was attributed to

an increase in the singlet-triplet transition probability from increased

spin-orbit coupling due to an external heavy atom effect. The increase

in spin-orbit coupling was later proved by McGlynn et al. (96).

A spin-orbit coupling increase was the reason given by McClure

(97) and Gilmore et al. (98) for the internal heavy atom effect. Transi-

tions between states of different multiplicities are forbidden due to

the selection rule requiring conservation of spin angular momentum. It

is never really possible to have pure spin states because the spinning

electron has a magnetic moment which can interact with the magnetic

field associated with orbital angular momentum (an electron moving in

the electric field of the nucleus generates a magnetic field). Because





-93-


of the interaction of these two magnetic fields, it is only possible to

conserve total angular momentum rather than spin or orbital angular

momentum independently. The mixing of states of different multiplicities

(singlet and triplet) is proportional to the spin-orbit interaction

energy and inversely proportional to the energy difference between the

states being mixed (99). The spin-orbit interaction energy for a
4
hydrogen-like atom is proportional to Z where Z is the atomic number.

This Z' dependence is the origin of the term "heavy atom effect" (100).

A major point of discussion is the nature of the state mixed with

the emitting triplet. Three types of states have been proposed to mix

with the lowest triplet to increase the transition probability, which

are

(i) the transition from the triplet to the ground state in

molecule, M, mixes with a charge-transfer transition in a

charge-transfer complex, MP, where M is an electron donor and

P, the perturber. is a heavy atom containing electron

acceptor (101);

(ii) the triplet-singlet transition in molecule M may mix with an

"atomic like" transition in the heavy atom containing per-

turber, P (102);

(iii) the triplet-singlet transition in molecule M mixes more

strongly with an allowed transition in molecule 1 caused by

the perturbing species, P (103).

There seems to be fairly good agreement that the charge-transfer

mechanism (i) or exchange mechanism (ii) is the most important. Some

investigators (100,104-106) favor a charge-transfer mechanism while

others support the exchange mechanism (89,107-112). There is excellent




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