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Optimization of signal-to-noise ratios in analytical spectrometry

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Title:
Optimization of signal-to-noise ratios in analytical spectrometry the external heavy atom effect in pulsed laser time resolved phosphorimetry
Added title page title:
The external heavy atom effect in pulsed laser time resolved phosphorimetry
Creator:
Boutilier, Glenn David, 1953- ( Dissertant )
Winefordner, James D. ( Thesis advisor )
Bates, Roger G. ( Reviewer )
Schmid, Gerhard F. ( Reviewer )
Li, K. P. ( Reviewer )
Moye, H. A. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1978
Language:
English
Physical Description:
vi, 186 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Atoms ( jstor )
Background noise ( jstor )
Ions ( jstor )
Lasers ( jstor )
Luminescence ( jstor )
Molecules ( jstor )
Noise measurement ( jstor )
Phosphorescence ( jstor )
Signals ( jstor )
White noise ( jstor )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Spectrum analysis -- Research ( lcsh )
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.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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05320952 ( OCLC )
AAK0661 ( NOTIS )

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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




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

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UNIVERSITY OF FLORIDA iimmm 3 1262 08552 5714

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ACKNOWLEDGEMENTS 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. n

PAGE 4

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 1 n

PAGE 5

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 IV

PAGE 6

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 signal-to-noise ratios of paired readings is given for additive and multiplicative noise using the relation between the autocorrelation function and the spectral noise pov/er. 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 high (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.

PAGE 7

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. VI

PAGE 8

CHAPTER I INTRODUCTION 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/N 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 (multiplicative noise) will be derived for several different measurement arrangements and optimization of S/N ratios will be discussed. General signal expressions in analytical spectrometry wil 1 be given along with S/N ratios for analytically important situations in emission and luminescence spectrometry. The generally useful S/N 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

PAGE 9

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 10fold 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 excitation 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 betv/een the phosphorescence lifetime of a given molecule and the lifetimes of other sources of interference such as stray light, fluorescence, or phosphorescence from the solvent. Aaron and Winefordner (14) have reviewed the available techniques in phosphorimetry along with their analytical

PAGE 10

3applications. 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/N 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 phosphorimetry. 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.

PAGE 11

CHAPTER II SIGNAL-TO-NOISE RATIOS IN ANALYTICAL SPECTROMETRY Noise and Signa1-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 predictable 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. Inserting the specific time response and frequency response of the

PAGE 12

-5measuring 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 subtraction, 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 signalto-noise ratio are useful parameters to describe figures of merit of analytical 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 CL(or q^) ^-^^-^ = -^ (II.l) which ties together two of the analytical figures of merit, namely the limit of detection (concentration, C. , or amount, q. ) and the sensitivity, S. The limit of detection is also related to the blank noise level, 0.,, resulting from 16 measurements of the blank where X,, is

PAGE 13

6the average blank, o, 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) K'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 + aX 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 normalized so that / f(X)dX = 1. Ensemble averaging of a function g(X) — CO is defined as g(X) = / 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(^) -=^o-^'Uj ni.2) J where e is the elementary charge, C, and i. is the j-th component in the current, A. The spectral noise power considered as a function of 2 frequency, f, is called the noise spectrum. The units of S. are A s and bars denote average values. Excess low-frequency noise has a noise power spectrum which increases towards low frequencies and has a frequency dependence often given by f " where a is close to unity (flicker noise). In spectrometry, 1/f noise is the most common and so will be the only one discussed in

PAGE 14

-7detail. 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 production, and detection. The spectral noise power in terms of current fluctuations for flicker noise is given by ^2 (S.)^^(f) = ifj] (II. 3) 2 where f is the frequency, Kc^ is a constant with dimensions unity which describes the low-frequency stability of the noise source and i. is as J —2 defined previously. We note that the flicker noise power varies as iwhereas the shot noise power varies as i,-; 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. 1 .f . noise. When combining noises of different origins into a total noise expression, the method of addition must be carefully considered. For

PAGE 15

-8example, if two noises with r.m.s. -values o and o, exist together, the r.m.s. -value of the total noise, Oj, is given by °T ~rl ^ % ^ 2%% (i^-^a) where c is a correlation coefficient; |c| ranges between |c| =1, in the case of complete statistical correlation, and c = 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 = a^ = n (II. ^b) where o is the standard deviation of n. 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

PAGE 16

-9with advantage make use of the relation between the auto-corre1at1on function and the spectral noise power involved. The auto-correlation function of a continuously fluctuating signal dx(t) is given by ^^(t) = dx(t)dx(t + t) (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) = 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 constant light signal), ii; (t) differs from zero only for t = 0, i.e., A \ij ii) = for T / 0. The values of dx{t) at different times t are A 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 x, i.e., ij^ (x) differs from zero also for x / 0. A Statistical correlation for x / also occurs when shot noise is amplified 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 S (f) through a A Fourier transformation:

PAGE 17

•10S (f) = 4 / dx(t) dx(t + t) cos(ajT) dx (II. 6) 4 / ijj (t) COS(cot) dr ^ and i>J^) ^ / SJf) COs(a)T) df (II. 7) X 5 X with 0) = Zirf. The Fourier transform of a delta-function, which describes ij/ (t) A 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 statistical properties will generally be changed. When a meter with time constant x 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 X / 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 x-, a correlation will exist between the results of two integrations when they are taken less than x. seconds apart. When they are taken more than x. seconds apart, the results are again strictly uncorrelated. For e.l.f. noise, a similar reasoning holds, i.e., an extra correlation is introduced in the noise signal when the integrator readings are taken less than x^. seconds apart; when the readings are taken more than x. seconds

PAGE 18

-11apart, 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 background fluctuations, one usually applies paired readings . The background, which has been admitted to the measuring system during a time long compared 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 x seconds later; x is called the sampling time. This difference. Ax, is taken to be the signal reading corrected for background where Equation II. 8 can be rewritten as ^^ = ^s^^o "" "^s^^ t^^b^^o ^ 's^ " ^^b^^o^^ (^^-^^ where dx, (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 + x ), divided by the standard deviation o , in the difference of the background flucuations occurring X seconds apart (see Figure 1). We assume the noise in the signal to be insignificant as compared to the background noise, and so c X (t + X ) Ax

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Figure 1. Representation of Signal and Noise Measured with a Meter a-j Signal Photocurrent, is, ys^ Time and a2 Fluctuating Background Photocurrent, ib, v£ Time, b] Meter Deflection for Signal, Xg, v£ Time and b2 Meter Deflection for Background, x^,, vs Time. KEY TO SYMBOLS : is signal primary photocurrent j_b = background primary photocurrent ib = average background photocurrent Xs signal meter deflection x^ = background meter deflection x^b = average background deflection t = time tg = sample producing signal introduced Ts = sampling time T(= time constant of meter damped by RC-filter Tp = response time of meter deflection _ dxbCto) = fluctuation in background deflection from X5 at tg dxbltQ+Tg) = fluctuation in background deflection from x^^ at tg

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•13t.. is a, a?. ci -^--^^V'^^^^/AcAA'"'^^^^ .aMW MH W -:ctk.:tt ' atfgQurji^i wt:_|iiw.^ry.-aMaigf' «'-g.-t5^aiPiiyi ju w ' wvg
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14with 2 From Eq. 11.11, the variance o can be straightforwardly expressed as o^ = dx. (t +tJ^ + dx, (t )^ 2dx, (t +Tjdx, (t ) (11.12) Ax b' s b^ b^ s' b^ o' ^ 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 a, which is the time-independent variance of dx, (t). From the 2 very definition of the auto-correlation function, a may be rewritten as o.„ = 2of; 2dx,(t +Tjdx,(t ) = 2[^ (0) * (t )] (11.13) Ax b b^ s b^ * X ' X s ^ ' where and '^(0) dXb(to ^/ dx,(t^)2 = ol ^(^) ' dx,(t^ + ^)dx^(t^) 2 To calculate a , the auto-correlation function is expressed in A A terms of the spectral noise power S. (f) of the background current ^b fluctuations and in the characteristics of the measuring system, using the Wiener-Khintchine theorem. Therefore, ij; (t ) may be expressed as A J oo Ki^J / S (f)cos(27TfT Jdf (11.14) A o *-» A O where S (f) = S (f)|G(f)|2 (11.15) ^ ^b and G(f) is the frequency response of the (linear) measuring-readout

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•15system. In other words, the spectral noise power of the meter fluctuations is the product of the spectral noise power of the background current fluctuations, S. , and the squared absolute value of the fre'b 2 quency response of the measuring system, |G(f)| , including the amplification 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 ^ (t J / S (f)|G(f)|^cos(2TTfTjdf (11.16) ^ ^ b ^ 2 Using Eq. 11.16, Eq. 11.13 for a may be rewritten as AX al= 2 j S. (f) |G(f) |^{1 cos(27TfT )}df (11.17) AX Q 1^ S 2 because cos(2TTfT^) = 1 for x = 0; o. is therefore a function of the ^ s' S AX sampling time x and as x -> both a and x approach zero. It 2 should be noticed that the factor 1 cosZrrfx (= 2sin Tifx ) stems from s s the use of paired readings. The noise components having frequencies f for which fx = 1, 2, 3, etc. are completely rejected. The signal deflection, x (t + x ), due to a constant signal current ^ SOS i that is instantaneously applied to the input at time t is x^tt^^x^) ^Gi^ x(x^) (11.18) where G is the d.c. response of the detector plus measuring system, and x(x ) is the normalized time response of the system used (meter or integrator), to a unit step function. Introducing the normalized

PAGE 23

-16frequency response of the measuring system, a(f) . G(f) , cm MT ^Q) 9(^^ GToy "G^ (11.19) Equation 11.10 for the signal-to-noise ratio finally becomes ^ = —^ (11.20) [2 / S. (f)|g(f)|^{l-cos(27TfT J}df]^/2 ^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/N ratio for specific situations, we have to introduce in Eq. 11.20: a. the background noise spectrum S^ (white noise or flicker noise) ; ^b b. the time response x(t ) 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 photomultiplier. All currents, i, refer to primary (or cathodic ) currents or count rates, respectively. An anodic current, i , is related to the a cathodic current, i , by i = i G f II 21 ) a c pm V 1 i . ^ I ; where G is the average gain of the photomultipl ier. This expression

PAGE 24

-17can be used if one wishes to convert final expressions for S/N 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 i, is assumed to be continuously present. The step response of a meter damped by an RCfilter (see Figure 1) or the normalized response of a meter when a constant d.c. current is suddenly applied at t = t , is x(t^) 1 exp(-T^/T^) (for T^ > 0) (11.22) where the meter time constant x = RC. The response time of the meter is defined as T^ = 2711^ (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)l^ = ^ 2 = ^ 2 (1^-24) 1 + (2TrT^f)'' 1 + (fx^)^ Inserting Eqs. 11.22, 11.23, and 11.24 in Eq. 11.20, with S. (f) for shot 'b noise, one obtains ^ i {1 exp(-27Tx /x )} S _ s '^ s r N |2 1 + f X S^{l-cos(27TfT^)} ^,2 (^^-^^^ {2 l-^_ Yl ^df}'/2 r The integral in Eq. 11.25 can be evaluated by using

PAGE 25

•18f sin X , I f^ -'^'^\ / -2-7-7 d^ = 4 (1 e ) TT + X which yields ^ i (1 exp(-2TTi /t J}""/^ | = _s_ s r (jj_26) For fixed x , the maximum value of S/N is reached for x^ = °° and is r s S _ ""s _ ""s , 1/2 ,,, „7V N , . , J/2 ,„ ^j J/2 V Ui.^/J Since the value is reached within 0.2% for x = x , the sampling time x S r r 3 5 can be restricted to that value, A larger value of x is only a waste of time; a smaller value yields a smaller S/N ratio. Equation 11.27 shows that the S/N ratio is proportional to the square root of x and thus improves with increasing response time x , provided x < x . D.C. Measurements in the Presence of Background Flicker Noise 2_? Substitution of the spectral noise S. (f) = Kfiu/f into Eq. 11.20 b yields i^[l exp(-2TTx^/x^)] p o °° 1 cos(27ifx ) 77^ ^ ^ f(l + f^l) ' (11.28) This expression is valid for any x and x , but can be evaluated only by numerical methods. It is possible to simplify this expression by introducing two new variables with dimension unity. Let B and z be defined as

PAGE 26

CO. o o o u o C3>

PAGE 27

20O

PAGE 28

-21 3 E 2tit^/t^ = t^/t^ (11.29) z = 2^fT^ (11.30) Substituting these new variables into Eq. 11.28 leads to „ i {1 exp(-3)} rr—z^ (11-31) ;9i/2-^2 r 1 cos z ,1/2 ' % z(l + zVb^) or ^ ^ f(B) (11.32) ^ (2k2t2)^/2 where f(e) ^-II^^jxeLiM] (n.33) f ; 1 cos Z dZ il/2 z(l + z7g^) Numerical evaluation (24) of f(6) gives a maximum of approximately 0.88 at B approximately equal to 0.8, i.e., x ^ t /8 or x R; 0.8 x , and f(6) falls to zero as B tends toward zero or infinity. A plot of f(B) vs B is given in Figure 2. The important point is thus that the maximum S/N for flicker noise is dependent only on the ratio x /x and not on x and x individually, and s r s r so there is no gain in S/N here when we make x (= 8x ) larger. Evidently in the flicker noise limited case, the increased smoothing effect of a longer time constant x = 2tix is just offset by the increase in lowfrequency noise from the equally longer sampling time x , 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 x (and x ) is increased. r s' I

PAGE 29

22The optimum S/N for background flicker noise is therefore _ i i N" ^ 2-2 1/2 ~ (11.34) where e,. = 0.81 K-: 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 ,, 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 background shot noise. In Table II, the flicker factors, E,, 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

PAGE 30

-23i> OO a c: (O to S•a c: o ro o •r— > •a io c o •nl/l OJ SQ. X +->

PAGE 31

24Table II, Expressions for Flicker Factors, C, for Several Measurement Approaches Measurement Device d.c. a.c, Current Meter K. = Jo.65K^ = 0.81 K. ? = K^{TT/2f .)^^^ dm V f f am f mod Integrator ?^. e jlnZK^ 0.83 K^ c^. = y(2f^^^) 1 /2 Synchronous Counter — e . = K-{£n2/f ,) ' ^si f mod'

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25stationary fluctuation process. The background current, i, , v;as 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 yery nature of the noise source considered makes it impossible to ignore the noise in one of the measurements. Since these signals are read after a sampling time t which may be shorter than the response time, x, 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, i (t), and reference signal, i^(t), are noise-free; (ii) the time dependence of the input signal is a step function, i^{t) ig for T < t < T + T^, i (t) = i for < t < T , and i^(t) = i (t) = for t outside the given intervals; (iii) at t = and t = T, the meter deflection caused by the preceding signal has decayed (T ^ x ) or been reset to zero; (iv) no additive noises are present;

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26(v) i is proportional to the analyte sample concentration (C ) s ^ and i is proportional to a reference parameter (C ) 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) =^ + 7(11.35) c (ix) the estimate of the analyte concentration, C , is given by X (T + T ) C ^-7 — r^C (11.36) Several points should be carefully noted. The noise in the multiplicative factor, G(t), is itself a stationary noise process, but x(t) is not a stationary noise process. The reference signal, i , and the reference parameter, C , 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 + T.) dx^d J -r^ = -^Tt-T— ^-^V^ (11.37)

PAGE 34

OJ QJ en •1 —

PAGE 35

28«^tK«

PAGE 36

29and the variance of C , op , is given by dx (T + T ) dx (t ) s^ s r s x (T + T ) X (t ) s s r s' (11.38) The relative variance of C may be written as dx (x )' r^ s' X (t )' r s dx (i r^

PAGE 37

30or (see Eq. 11.40) x^(t^) rjry + dx^d^) (11.43) and dx (x ) is given by ^s dx^d^) = i^exp(-T^/T^) J exp(u/T^)dG(u)du (11.44) From the previous evaluation of x (t ), the expression for the meter deflection due to the analyte signal is x (T + x ) = x (T + x ) + dx (T + x ), where Xgd + x^) = is^^^GCl exp(-x^/x^)] (11.45) and dXg(T + x^) = i^exp(-x^/x^) J exp[(v-T)/x^]dG(v)dv (11.46) where v is a dummy variable for integration. To find the expression for dx (x )dx (T+x ), Eq. 11.44 and Eq. 11.46 are multiplied and ensemble averaged. It is found that dx^(x^)dx^(T + x^) = i^i^exp(-2T^/x^). ^ •Jdu / exp[(u + VT)/x^]dG(u)dG(v)dv (11.47) 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 dx^(x^)dx^(T + T^) = i^i^exp(-2x^/x^)^s T+xs / du / dv exp[(u + VT)/x ]dG(u)dG(v) (11.48)

PAGE 38

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 ij;g(s) = 'dG(t)dG(t + s) (II. 4S) The term dG(u)dG(v) is therefore equal to i|jp(v u). Rearranging Eq. 11.48 and replacing the integration over v by y = v u for given u results in dx (t )dx (T + T ) = i i exp(-2T /t ) r s s s r s ^ s c T T-U+Tg / du exp[(2uT)/t^] / exp(y/T^)^p(y)dy T-u (11.50) This is the general expression for dx (x )dx (T + t ). 3 K r s s s In an entirely analogous fashion to that in which the expression >2 . for dx (t )dx (T + t ) was obtained, the expression for dx (t) is found to be dx (t ) = i exp(-2T /t ) r s r s c S -U+T^ / du exp(2u/T ) J exp(s/T^).j;^(s)ds -u c'^G' (11.51) where s = u' u for constant u. Substituting Eqs. 11.40, 11.50, and 11.51 into Eq. 11.39, the expression for the relative variance of C is s c C 2exp(-2T^/T^) C^ T^G^Ll exp(-T^/T^)]'^ -U + T ( " '^S o\ /du exp(2u/T^) / exp(s/T^)i^p(s)ds + tI n C -^ C b ^ -U S T-u+T 1 / du exp[(2u-T)/T ] / ^ dy exp(y/r )4'r(y) (11.52) ^ T-u c G J

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-32The 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 a" 2exp(-2T /t ) / du exp(2u/T J _S ^ ^ ^ ^ C^ t•[1 exp(-T /t )f s c '^ s c / ^ ds exp(s/T^)[4.g(s) i|.g(s + T)]| (11.53) •U + T / -u From Eq. 11.14, the Wiener-Khinchine theorem, CO ,|; (s) t|. (s + T) = 2/ S.(f)sin^f(2s + T)sin^fTdf (11.54) Substituting Eq. 11.54 into 11.53, gives the final, general expression for the relative variance of C , which is Eq. 11.55. al 4exp(-2T /t ) / du exp(2u/T ) ^s ^ ^ ^ ^ ^ ?~ t•[1 exp(-yT^)]2 -U + T / ds exp(s/T )• -u ^ oo • / S (f)sinuf(2s + Dsin^fTdf (11.55) ^ The integral over u is defined over the range < u < t ^ T, and the integral over s is defined over the range -u < s < -u + t. Up until this point, the derivation of the expression for the relative variance of C^ was general for Sg(f), x , x , and T subject to the constraints of the assumptions. The divergency of flicker noise as f ^is neutralized by the two sine functions of frequency, f, in

PAGE 40

33Eq. 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, t of noise dG{t) by ' 4'g(s)ds where (|jp(0) = dG(t) . Because i|>p(s) differs from zero only for |s| ^ ip, while Tg < T , T , and T, for this case, SJf) is a constant over the relative frequency range, but falls off at ZTrf ^ t~ . Starting from Eq. 11.53, i|^g(s + T) = because (s + T) >> t^. Because 4)As) exists only for s % 0, the integral over s can be approximated by / tj;p(s)ds. — oo It is a valid approximation as ;i u < t ; s is within the integration limits of -u and -u + t . From Eq. 11.56, the definition of i|;p(s), and the approximation of the integral over s, Eq. 11.53 becomes ol 2exp(-2T /t ) / exp(2u/T )du dG^ t^ ^^5. S C Q C b ~r— 2=? ? iu.57) C3 tV[1 exp(-T^/T^)]'^ Making the substitution z = 2u/t and evaluating Eq. 11.57 gives 'C 2 dG'^ T [1 exp(-2T /tJ] c; GS [1 exp(-T /t )]^ 2 From the definition, dG = ti^gCO), the inverse Wiener-Khinchine theorem,

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-34and Eq. 11.56 (see Eq. II. 6) dG^ = Sg(0)/4Tg (11.59) Substituting Eq. 11.59 into Eq. 11.58 yields °C S (0)[1 exp(-2T /t J] V= -=? —^ {u.eo) C^ 2GSJ1 exp(-T^/T^)]'^ The S/N ratio is therefore ,Gv^ [1 exp(-T /t )] jl^ , .^. . . -(n.61) ^ \(0)l^ exp(-2T^/T^)] 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 = 2ttt where 2TrT 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 N "lax = ^ (11.62) 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 /T". r It should be noted that the expression for shot noise may not be substituted here for Sg(0) because shot noise is not a multiplicative noise. All that can be specified is that for the white noise case S^(0) is constant. The S/N ratio will also increase as /S^(0) decreases.

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35D.C. Measurement with an Integrator for White Noise The case of an integrator may be derived from Eq. 11.61 by taking the limit as x -^ °° for an integration time x. = x (28). The result for the case of white noise is given by G J7' S ^ ^i v^^ (11.63) This shows an improvement in S/N ratio over a d.c. meter by a factor of /n" assuming x. = x . 3 1 r D.C. Measurement with an Integrator for Flicker Noise It is necessary to assume that x< x , as was the case for the integrator in the case of white noise. Starting from Eq. 11.55, 2 setting Sp(f) = K^/f for flicker noise, and approximating exp(2u/x ), exp(-2x^/x^), and exp(s/X|^) by unity give 7^ = J{l(7:-^)'^"(T-i)^i(^^l)'^"(^^^i)(-^)^£nT £nx.| (11.64) With a fixed integration time x., the minimum value of T is given by T . = X(see assumptions). Solving for the S/N ratio gives mi n 1 ^ ^ I (T^x ) = — ^ (11.65) '^ ^ 2K An4 If T is increased relative to x . , for the limit of T >> x., the 1

PAGE 43

-36signal-to-noise ratio becomes I (T » T.) = ^ , — (11.66) ^ ' 2K^/2 + £n(T/T.) As T increases, the S/N ratio decreases. For a fixed total measurement time, the optimum S/N will be achieved by making n measurements of reference and standard vn'th 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 measurements 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 , x , and T without numerical integration. If one assumes T >> T , 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.

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-37Signal-to-Noise 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 absorption 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 dependent, 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

PAGE 45

-38measurement system. In the following treatment, flicker noises will be assumed to be completely dependent or completely independent (no correlation 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 expressions would be exceedingly complex and of little use since correlation 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 "background" 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 spectrometry, 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

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-39or luminescence and continuous measurement), AM (amplitude modulation of emitting radiation in emission spectrometry or of exciting source in luminescence spectrometry), WM (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), AM + 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 WM + 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 C is given by the appropriate AC-expression, i.e., E,^for the synchronous counter, rather than by the d.c. integrator a 1 expression, £...{£... > E, .). ^ di di ^ai In Table IV, the appropriate flicker factor, £,,. or i . 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

PAGE 47

40source 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, D „ and U ^ replace the values of 1/2 or 1 in Table V for CW, AM, WM, SM, AM + WM, and AM + SM measurement 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 CW 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 e\/ery 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 + WM 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

PAGE 48

-41appropriate expression from Table I vnth 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 D^-m. D, „, and D„p, EM LM 6D are those in Table V, and the ones for D^^, D ., and D--^ are given in Table III. 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 SM, 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, continuum scatter or molecular band interferents in atomic fluorescence flame spectrometry, etc., will be reduced as the modulation frequency, f ., increases and the measurement system's noise bandwidth, Af, decreases, i.e., the flicker factor, c is related to Af/f , by ai mod ^ ^ai»F== IK =11 (".67) ' where t 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 ^,. decreases inversely with /n. There is a practical a I limit to f ^ . and therefore to Af/f ., namely, the time to mechanically mod mod -^ change from sample to blank with no memory effects, and so f , < 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

PAGE 49

-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. , ^ .<£;,. . If a 1 di the samples and standards are identical in all respects, except for the analyte, then WM corrects the signal level for unmodulated signal components and reduces flicker noises due to these sources. Because WM 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 c • a/Af/f T as in Eq. (11.67), e . can be made ai mod an 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. Cone! usions 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 multiplicative, the S/N ratio increases as the square root of the

PAGE 50

-43response time, x , or the integration time, x., for current meters and integrators respectively; (ii) For background shot noise limited cases, modulation techniques will give S/N ratios ^ 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 multiplicative, 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 multiplicative, the S/N ratio is approximately independent of response time or integration time; (v) For the cases of flicker noise, whether additive or multiplicative, the S/N will decrease with increasing sampling time relative to a fixed response time. It is optimum to make 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, ito 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/f , as small as possible (either with small Af or large f .);

PAGE 51

-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 simultaneously. In some situations, an internal standard, excitation source intensity, etc., measurement may be made simultaneously 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.

PAGE 52

45Table III. General Signal-to-Noise Ratio Expressions for Emission and Luminescence Spectrometry with Definition of Terms S N 1^2 + m2 ^ ^,2 +^^ +(^ ^0% ^^q^, \2 , ,^w„ ^2 , ,^w„ x2 fe ' SS '^^IS'SS' ^\f ' ^^^ F ' ^\f^ ' (2-^Df) ' (2\)' C fc; Measurement Mode w CW AM WM SM 1

PAGE 53

-46Table III. (continued) Definition of Terms hs = analyte emission shot noise = J^ DrMD, ,MRrt , counts V ^ EM WM E m Ndc = background emission shot noise = ./DZTdTrTF", counts BS ^ V EM B m N, c = interferent (in matrix emission) shot noise = ./dZTrTT", counts lo u EM I m P » o N e DS detector dark shot noise = v/DTTRTt", counts V CjD D m' N, c = analyte luminescence shot noise = J-fy D, ^D, ,pu,Dr,R, t , counts Lb V ^ LM VJM L m N^^ = scatter (source) shot noise = JD. j^D „D^R^t , counts N, ^ = interferent (in sample/blank) luminescence shot noise = >w .[dTT^DTrTT", counts V LM I^ m Z^'N. = amplifier readout noise (generally negligible in S/N measurements), counts Ncr = analyte emission flicker noise = -^ Cri-DrMD,,MDnRr-t , counts tt2 ^EF EM WM E m 2^Ngp = background emission flicker noise = ^'^^rF^Dm'^SB^O^B^ ' ^°^"t2 2%, c = interferent (in emission flicker noise = 2^r, r-Dr-MRi t , counts i„r ^I F EM I m e e e Z^Nnc = detector dark flicker noise = 2^Cr,cD^„Rr,t , counts Ur ^UlbD D m z'^NgP = scatter (source) flicker noise = 2'"c3pD^f^D|^^D^gDQR2t^, counts 2 Nt p = interferent (in sample/blank) luminescence flicker noise = 2 Ct rD. mDoRt t , counts I^F LM I^ m N, p = analyte luminescence flicker noise = -^ e„D, ^D^r,D„R,t , counts 1-1 £: L Ln OD (J L m Sp = analyte emission signal = -p^D^.,D, ,MRr-t , counts t ^2 EM WM E m S. ^ analyte luminescence signal = -i Di mD, ,.,Dr,R, t , counts L ^ ^ LM VJM (J L m t = measurement time for one spectral component, s (see Figure 3 and text) Dj^l^ = amplitude modulation factor for luminescence spectrometry, dimensionless

PAGE 54

47Table III. (continued) Definition of Terms (continued) Dpn, = emission modulation factor for emission spectrometry, dimensionless DcD sample-blank factor, fraction of time sample is "on," dimensionless '^WM ~ wavelength modulation factor, dimensionless D 1^ = wavelength modulation factor for narrow line, dimensionless D^ = factor for correction for emission in luminescence spectrometry, fraction of time emission or luminescence (equal times) is measured, dimensionless Dpp, gated detector factor to account for fraction of time detector is gated "on," dimensionless Rp = photoelectron counting rate of analyte emission, s Rn = photoelectron counting rate of background emission, s b Ry = photoelectron counting rate of interferent in emission spec^ -1 trometry, assumed to be in both blank and sample, s R<~ = photoelectron counting rate of source scatter in luminescence spectrometry, s R. = photoelectron counting rate of interferent luminescence in luminescence spectrometry, assumed to be in sample and blank, s R^ = detector dark counting rate of detector, s R. = photoelectron counting rate of analyte luminescence, s~ 5 P = flicker factor for analyte emission flicker, dimensionless 5t p = flicker factor for emission interferent flicker factor, e dimensionless

PAGE 55

-48Table III. (continued) Definition of Terms (continued) Cnp = flicker factor for background emission flicker factor, dimensionless Cc-n = flicker factor for source scatter (in luminescence spectrometry) flicker factor, dimensionless 5y p = flicker factor for luminescence interferent (in luminescence ^f spectrometry) flicker factor, dimensionless Cpip = detector flicker factor, dimensionless Ci p = flicker factor for analyte luminescence, dimensionless

PAGE 56

-49Table IV. Evaluation of Flicker Factors in Emission and Luminescence Spectrometry EMISSION^ Measurement Mode

PAGE 57

50Table V. Evaluation of Duty Factors in Emission and Luminescence Spectrometry EMISSION* Measurement Mode TM D WM CW AM WM SM # # 1 1/2(1) 1 1 # 1 1 1/2 1 GD LUMINESCENCE* Measurement Mode LM WM D SB m D. GD CW AM WM # # SM AM + WM AM + WM 1

PAGE 58

51 Table V. (continued) Notes (continued): D|_f,i] = 1 if the source of excitation in luminescence spectrometry is not modulated D^g = 1/2 for paired sample-blank measurements D5B = 1 for sample modulation DgD = 1 if the detector is "on" during the entire measurement Dqq < 1 if the detector is gated ^iH ~ ^ ^^ ^^^ exciting source in atomic fluorescence spectrometry is a continuum source D^l/\ = 1/2 if the exciting source in luminescence spectrometry is a line source Dq = 1 if the analyte emission in luminescence spectrometry is automatically compensated for as in AM Dq = 1/2 if a separate "source off" measurement must be made in luminescence spectrometry to compensate for analyte emission as in CW, WM, 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 AM mode where DrM and D, ^, are both as shown in parentheses. Line means a line interferent; cont means a continuum interferent.

PAGE 59

-52Table VI. Duty Factors for Pulsed Source-Gated Detector Cases Pulsed Source-Gated Detector^— No Time Resolution (No Delay Between Pulsing and Detection) -t /t. {t T.[l e 9 ^]} g dnn ~ t /t GD g' g Pulsed Source-Gated Detector^--With Time Resolution (Delay of t., s Between Pulsing and Detection) -t_/T. -t /t. -t./x. T.[l e P ^][1 e 9 ^]e ^ ' tg n e ^] dpn = t /t GD g' g Definition of Terms t = pulse width of source (assuming rectangular pulse), s t = gate width of detector (assuming rectangular gate), s tj = delay time between end of excitation and beginning of measurement f = repetition rate of source (gate), Hz T^. = lifetime of radiative process, i, s a The duty factors, d^M or d^jy,, become dgQ in the event the radiative process, i, is not pulsed. These expressions apply to an averager; one must replace tg in the denominator by 1/f for an integrator.

PAGE 60

CHAPTER III MOLECULAR LUMINESCENCE RADIANCE EXPRESSIONS ASSUMING NARROW BAND EXCITATION Assumptions In the derivations to follow, it v/ill be assumed: (i) that all molecules are in the condensed phase at room temperature 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 bandwidth; (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-

PAGE 61

54E(v„ ) i-^^ / B„ a„(v,v )G(v,v )d^ (III.1) and the single line de-excitation rate for stimulated emission is given by ^(".,.' / B a (v,v )G(v,v )d\ •' M,i6 IJ (III. 2) -2 where E(v ) is the integrated source irradiance, Wm , c is the velocity of light, ms~ , B„ and B „ are the Einstein coefficients l,u P,£ -13 -1 for absorption and stimulated emission respectively, J m Hz s , a„(v,v ) anda (v,v ) are the normalized spectral profiles of the lower £^0 y and upper levels respectively, Hz' , and G(v,v ) is the normalized spectral profile of the excitation source. For molecules in the condensed 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 solvent molecules (37). If the only broadening present is assumed to be homogeneous broadening, then the normalized spectral profiles are given by "£(^'^0^^%^^'^0^="(^'^0^=1 6v/2 l_(v-vJ^ + (6v/2)^_ (III. 3) where 6v is the absorption bandwidth and v is the center frequency. If the excitation source profile G(v,v ) is much narrower than the normalized absorption spectral profile and the source is operating at the line center, v , then a(v,v^) tt5v (in. 4)

PAGE 62

-55The excitation and de-excitation rate therefore become, respectively, and -3 -1 where p is the spectral radiant energy density, J m Hz , and E=E(v ). 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 transitions is required. The absorption rate is then given by |^Y^/\„.".,i(^'%)G(^'^o)d^ niI-7) 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. i

PAGE 63

-56The energy efficiency for such a process is given by (III. 8) '21 '2j,10^{ ^20, li ^ "^21^ and the quantum efficiency I ^20, li 21 ?A20,li '^ (III. 9) 21 where A^Pi 1 • = Einstein transition probability for emission (luminescence transition from the zeroth vibrational level of the radiatively-excited, 2, electronic state to the ith vibrational level of the lower, 1, electronic state), s ; kpi = nonradiative first order de-excitation rate constant for same transition given in definition of A^q -i -j > s ; v„» ,. = frequency of luminescence transition, Hz; ^2i 10 ~ ^"^^"^i^sncy 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 2j,10 , is given (38) by / ^ dv = (''2j,10 10, 2j ] B 10, 2j "1 V2 92"l (III. 10) where

PAGE 64

•57hv,^ p. = energy of absorption transition, J; c = speed of light, ms ; 3 -1 -2 B-iQ 2-j ~ Einstein coefficient of induced absorption, m J s ; g, = statistical weight of electronic state, k-, _3 n. = concentration of electronic state, k, m . The Einstein coefficients are related to each other (38,39) by ^20.1i ° [ j ' ) ^20.11 <""' where 3 -1 -2 ^?n li ~ Einstein coefficient of induced emission, m 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 «2o.ii= (fr2)(5:)s2o,„= (7T2'(i"^2l''!'«2o(«'l^,i('!) > ,2 V 2 ' 3^0^ ^ (III. 12) Bio,2j= (7V)(i-)^10,2j= (f7H^)(R^l)^ll^ -^0 ' -^0 ' (II 1.13) where 2 2 S„Q,li, S-jQ 2-j ~ electric dipole line strength, C m ; -12 2 2-1 c = permittivity of vacuum, 8.854 x 10 C (Nm ) ; -?4 -1 h = Planck's constant, 6.626 x 10 J s ; el 2 ? 2 (Rp-,) = pure electronic transition moment, C m ; |<6(Q) |e(Q)>| = vibrational overlap integral (Franck-Condon factor) between vibrational levels in two electronic states

PAGE 65

58involved 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, n^/n, , is given by ' ( '^ZCli ^2j,10 ,'°'^-J + k., for steady state conditions and for the condition of negligible thermal excitation (k^^ ^ 0)I" Eq. III. 14, E(v^q ^-j) ^s the source irradiance (integrated spectral irradiance) of the exciting line and 6v 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 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 III. 14 can be rewritten in terms of the quantum efficiency (see Eq. 1 1. 9). ^ t ^20,1 i ^ri N c\ ^ 7iC6v By utilizing the definitions of the A's and B's (see Eqs. I II. 11III. 13)

PAGE 66

-592 3 I ^20. 1i ^2j,10 qilS-,^,,) l|^ (III. 17) where all terms have been previously defined. If we now use the following substitutions for simplicity ^20, li -\<^2o^^)KiQ)\' ll^ A = A 21 2jJ0 ^21 ^2j,10 ^12 = ^10, 2j ' ^(^10.2j) then ^^12^2j,10^ ^2r2j,10 n^ "^^21^^ I ^20,li^20.1i "' , , ^^21^2.1,10^ ^2l4lJ0 (III. 18)

PAGE 67

60Simplifying Eq. III. 18 by use of the relationships between B,^ and B^, (Bp,gp = B?i9i) ^'^^ dividing numerator and denominator by V^ . ,^ gives V2 92"l r2EY^^ TTCApi 6v 3 '2J.10 I ^ZOJi^ZOJi 2j,10 ^^^21^21 TrcA„, 6v 2j,10 M 20,li'^20,li (III. 19) According to Strickler and Berg (39), I ^20,li''20,li / F(v)dv <^-L^>AV-^ (III. 20) 3 -1 where F(v) is the luminescence profile function and ~ is the reciprocal of the average value of v. in the luminescence spectrum. Because I V^q -.^ = 1, i.e., orthonormal complete set, Eq. III. 19 can be 1 rewritten as 92"l f^^VlVtl 3 -3 [7TcA2^6v J''2j,10''\ ""AV 2j,10 ^^^21^21 TTCAp,-] 6v ^2j,10"\ 'AV (III. 21) If as in atomic fluorescence (4), E*, a modified saturation spectral irradiance, i.e., E* is related to E^, is defined as V V cA E* V 21 12 ^21^21 (III. 22a) and if Co,-in is defined as = 3 -3 u =2j,10 ~ ''2j,10^\ ^AV^2j,10 (III. 22b)

PAGE 68

-61 then 2E ^2 1 I^12__ ^ 2E_ ^2j,10 "^" The fluorescence radiance expression (4,40) for a two level system is given by Be = (t-)Y (-^)n.(-^^^i^)B,„[l ^Ll] (III. 24) v/here £ is the fluorescence path length in the direction of the detector. Substituting into Eq. III. 24 for the ratio g-jn^/gpn, from Eq. III. 23 and for n-j in terms of n„ from Eq. 1 1 1. 23 gives B = (^)Y„ E* (^)( jf ^ ^ ) (III. 25) F 4. P2^ v^2 ^ 92^2j,10 By evaluation of ] A„„ ., . (combining Eqs. III. 16 and III. 20) ^^2oii=r^^ (^^^-26) i "^^''^ ^2j,10 and by substituting for Y in terms of Y21 (Eq. III. 8) and for E* (Eq. III. 22a) into Eq. III. 25, Bp becomes ^F = (fc'"2 p20,liS0,li (""' which is the expected expression based upon previous derivations for atomic fluorescence (4,41). However, it is interesting to stress that Bp is independent of the vibrational overlap integrals. Evaluating n^ in terms of n^, where n^ = n, + n^ ^ total concen^2 T T 1 2 tration of molecules in all electronic states gives

PAGE 69

628F=(|^'l*20,1iSo,li(J|-)f n. (III. 28) 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 radiatively excited and assuming the nonradiational excitation rate constants, ki2 and k-.^ and the radiational rate constant A^^ ^"^^ 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 , njn, , is n, TTc6v ^ ^20, li ^ "^23 ^ ^2]^ 2B,. ,.E itc6v '-I ^^20,li^^ 23 + k 21 " '^23'^32 (1 11.29) where all terms have the same definitions given previously except the levels involved may differ and E = E(v^. ,p). The definition of the power, Y , and quantum, Y, efficiencies for electronic transitions 3 -> 1 according to Lipsett (42) and Forster (43) are

PAGE 70

63l '^30,li''30,li ^31 I ^30, li ^'^31 ^"^ k k •^23 32 32 I ^20, 1i ^'^21 ^•^23'^ ex (III. 30) I ^30,1 i 31 I ^30,1i^''3l'"42 k k 23 32 I ^20,1 i "^"^21 "^''^23 (III. 31) where v 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 'P2i ~~ "''\^ (III. 32) ^21 ^32^21 (III. 33) where x^o ^^ the crossing fraction (also termed quantum yield of intersystem crossing or triplet yield) and is given by ^32 ^32 [ '^30,li ^^^31 "^42" y ^ k k 23 32 (III. 34) + k + k 20, li 21 "^23 where y is the radiative power efficiency and y„, the radiative P2I ^' efficiency, given respectively by I ^20,1 i 20,1 i '21 [P20,li ^ ^21 ^ ^23] (III. 35) ex

PAGE 71

•64and u '21 20 , 1 i I ^20, li ^ 4l "^ "^23 (III. 36) for 2 -> 1 luminescence excited indirectly. Combining Eqs. III. 29 and 1 1 1. 31 gives ^^3^31, 10 ^ ^1 ^r"^'" {^0,li ^^3j,10 ^ ^31 Trc6v I A3Q^^. (III. 37) + 1 Substituting for J A^q ^^ from Eq. 1 1 1. 16 (replace 2 by 3 in all terms) and making substitutions of B,^ = B,^ „• ^nd B-, = B». ,^, and A-, = A 3j,10 ^g3^31^1^^3.iJ0"3.iJ0 g^^cA3^6v I V3Q^^.v^Q^^. 1 + ?R Y FV 3 ' 31' 31 "3.i. 10^3.1, 10 (III. 38) ^"^^l^^^p30.1i^30,li)^ Using the Strickler and Berg (39) approach (see Eq. III. 20) and the definition of E* and C "13 3], 10 as cA E* V 31 13 ^31^31 (III.39a) f =3 -3 w ^3j,10 " ''3j,10^\ W3j,10 (III. 39b)

PAGE 72

65then n /n, is given by (!3, 2E 9l n6\) E* V (III. 40) 2E Tr6v f 13 3j,10 The fluorescence radiance for the 3 ^ 1 fluorescence transition is given (4,40) by ,2E 10^ ^F=(^)\(^^(-^r^)^--t^ V3 10, 3J'-' g3n^ ] (III. 41) Substituting for n-, in terms of n^ and for n^n, from Eq. III. 40 gives which is identical in form to the expression for the 2-level case (Eq. III. 27). Substituting for n-. in terms of n.|. (n-p = n, + n^ + n^) can be done using Eq. III. 40 for n-/n, and Eq. III. 32 below for n^/n. '32 I ^20,1 i ^ "^21 ^ "^23 (III. 43) and so E* V 13 2E 1 , ^^^^^3j,10 , ^3^ r 2E. '32 ^1 ^tt6v'^ I ^20, li "^ ^21 ^ "^23 (III. 44) and

PAGE 73

-66'f,.. -^h^ P30ji^^30ji 3-^1 1^3 f E* V 2E 7t6v C 13 1 + 3j,10 go 2E r 3'\ r _] ^32 1 ^20,1 i "^ '^21 ^ '^23 (III. 45) where the subscript on Bp 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 Br 2^1 1^3 f— )Y (hv 10. 3j )^10,3j"l^^ (III. 46) where Y is the quantum efficiency for luminescence from level 2 while P2I exciting level 3. Substituting Y (Eq. III. 32) and n, and n„/n, P21 I J 1 (Eq. III. 40) gives the expected relationship for 1 ^ 3 excitation 2-^1 W3 (^)n2 p20,li^"20,li (III. 47) and substituting for n„ in terms of n^ gives '^P2-.l ^ ^^^ p20Ji^"20.1i1-^3 1 + fp20.l1^^-*^"l ^21 23 ^32 E* 2E ^ "^13 T , ^^" ^3jJ0 r!3, ,1E_ (III. 48)

PAGE 74

67The final case of potential interest is radiative excitation of state 2 directly from state 1. In this case, Bp is given by ^2^1 ^ ^21 ^ U2 ^'^10,2j"l'-' " " J^^x,,-' g^n-j tt6v (III. 49) where E = ECv-.^ 2i)T*^^ ratios n^/n-, and n„/n, for this excitation case (2 becomes 3 and 3 becomes 2 in Eq. 1 1 1. 40) are 2 _ Itt) 2> 2E 'g-j'' tt6v E* V 2E ir6v ^ 12 2J.10 (III. 50) and '23 I ^30,1 i "^'^31^'^ 32 (III. 51) Substituting for n„/n, (Eq. III. 50), for n, (Eq. III. 51) and for Y (Eq. III. 32) gives 21 S-.1 ^ ^'^^"2 ] ^20,li^"20,li (III. 52) U2 and substituting for n^ in terms of n^ (n,. ^ n, + n^ + n^ using Eqs. III. 50 and III. 51)

PAGE 75

682-^1 1^2 ^4^^ [ ^20,li''20,1i E* 2E ^ "^12 (III. 53) n&v ' Co • T A 1 + 2jJ_p_ ^ ^23 f^Z^ r2E ^ ^30, 1i ^ 4l "^ ^2 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 Bp is Br(t~) y A„_ ,.hv_„ ,. Fo_).i ^^ i 30, It 30, n U3 n. 1 + ^^ ^30,li"^'^31 ^'^32 E* V 12 ^23 1 + 2E "^" " ^2.i,10 (III. 54) A nontrivial but analytically unimportant case is E-type delayed fluorescence, DF, where excitation of 3 from 1 occurs followed by intersystem 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 'P31 ^ '32^23^P3^ (III. 55) where Xoo is given by Eq. II 1. 34, and <__ and y are given (42) by ^32 32 23 '31 ^23 I ^20, li "" ''21 " '^23 (III. 56)

PAGE 76

69and '31 ^ ^30, li ^ '^31 "^ '^32^''3j,10 (III. 57) Substituting for Y into Eq. III. 41 and for n, and n_/n, as previously P3I I :s I for the case of 1 -> 3 excitation and 3 -> 1 fluorescence, gives i = (— ) •^23 32 ' l^[^0,li^^31^42^^[^20Ji^4l^43^, V 30, li 30, It '3 2E g-j tt6v T r ^3 93 Ui + ^ + -^ L 91 9] '32 [A + k +k ^J ^^ 20,1 i "^21^23 ^(^)^^ v' S 3j,10 (III. 58) where E H-^q^^^) Limiting Cases of Steady State Excitation In all cases given, high implies that E(v) >> E*6vir/2^ and low implies that E(v) << £*Svn/2E,. Limiting expressions are given for cases of analytical util ity. For a two level molecule, if the source irradiance is low, then Bp (see Eq. 1 1 1. 28) becomes 41?^ %)p20,li^^20,li(^)"T 92,„ f^^^" l0,2.1^^2j,10 l U2 E* Tr6v ^12 J (III. 59) and if the irradiance is high , then Bp (see Eq. III. 28) becomes ^F(^;)^(i)p2o,ii^^2o,ii^-A7^ (^^^-60) 1-^2 1 + 92

PAGE 77

70For a three level molecule assuming 1 -* 3 excitation and 3 -> 1 fluorescence, if the source irradiance is low , then Bp (see Eq. III. 45) becomes Bp(lo) 1-^3 •l^'^'ao.ii^-ao.iit^)"! ^^'" lo.a.i'^a.i.io E* tt6v ^13 (III. 61) and if the source irradiance is high (see Eq. III. 45), Br becomes 1*3 1 v^ ^ p20,li"^'^21^'^23 For a three level molecule, assuming 1 ->3 excitation and 2 -> 1 horescence 1 1 1. 48) becomes phosphorescence, if the source irradiance is Jow, then Bp (see Eq Bp(lo) ^2^1 1^3 (i'?'»20,li^"(5f) "1*^32 ?^20,li"^4l"^'^23' ' "13 ^^("I0,3.l)^3.i,10 l E* 7i6v (III. 63) and if the source irradiance is high (see Eq. Ill, 48), the Bp becomes Bp(hi) = (|-)yA,„ ,.hv„_ ,.f ^1^2 U 1 Ppi 4Tr 4 20, ll 20,ll yn , . j g 1^3 ' pO,li^21 '^23 1 +-i + ^32 ^ 1^20,1 i "^'^21 "^'^23 (III. 64) For a three level molecule assuming 1 -y 2 excitation and 2 -> 1 phosphorescence, if the source irradiance is Jow, then Bp (see Eq. III. 53) becomes

PAGE 78

-71Bp(lo) (fc) p20,liSo.li^4^"T ^^(" I0,2p^2j,10 E* •ii6v ^12 (III. 65) and if the source irradiance is high, then Bp (see Eq. III. 53) becomes Bp^^j) %) p20,lih-20,li U2 ^1 1 + -i+ ^23 '2 IA30^^.H3^H 32 (III. 66) 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 ^ ("I0,2j) '1 191 + g2J E* Tr6v "12 '2^2j,10'=\^W2j,10 (III. 67) For a 3-level molecule (1 -> 3 excitation), it is given by ^'^"10, 3j) = 193] 13 ^3j,10 n6v ^^"3j,10^\ ^AV k ' V^ 32 3 ?^20,li ^^21 ^^ 23 (III. 68) For a 3-level molecule (1 -> 2 excitation), it is given by ^'("10, 2j) .9; E* 12 1 '2j,10^ Tf6v '^''h,10'\"M ' V* ^23 2 l^0,li^^31^^ 32 (III. 69)

PAGE 79

-72However, Eq. III. 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. III. 67. For a typical organic molecule at 298°C, E* '\. 1.8 x 10"^ W cm'^Hz'^ 6 -2 -1 ''''2 (6 X 10 W cm nm ) (assuming Y„, = 1 and Xr,-. = 300 nm) or E* '\^ 1.8 X 10'^ W cm"^ Hz"^ (6 x lo'' W cm"^ nm"b (assuming Y^^ = 0.1 and 3 3 A21 = 300 nm). Assuming v^-10"^^ ^AV^2i 10 '^ ^ ' ^1 "" ^2 ^ ^ ^"^ 6v = 10 Hz (gaseous molecule) or 6v ^ 10 Hz (molecule in liquid s 2 solution), then E (v,q p-;) '^ ^0 kW/cm for the gaseous molecule and s 5 2 E (v-jQ 2.;) ^ 10 kW/cm for the molecule in the liquid state assuming Y^i = 1 and A21 = 300 nm. For a 3-level molecule, E^(vi^ ^.) will be c smaller than E (v^q ^0 ^^ ^ factor k^^/I A^g ^^ + k„, + k which will 57' ^ ' be 'v.lO -10 for most molecules (44,45). Nonsteady 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. 11.28, it is possible to define a steady state concentration of n^, "2 . This is given by "t "2ss = E^^ (III. 70) (1 + — )Cp,n OA + 10,2j 92 e2j,To

PAGE 80

73where the spectral radiant energy density, p, has been used. For a rectangular excitation pulse, p(t) = p for < t < t and p(t) = for t > t^ where t^ is the pulse width, s. The concentration of n^ as a function of time, n2(t), for < t ^ t is n2(t) = n^^gLl exp(-(a + bp^)t)] (III. 71) where /\ a = T^^—^ k 1 (III. 72) ^2j,10 ^' and b = B^2 "^ B^i (III. 73) For low irradiance cases, the growth of n^ population is controlled by the luminescence lifetime, a" . As the irradiance exceeds the saturation irradiance, the growth of x\^ population is more rapid. If the pulse width is long compared to the lifetime, the steady state concentration of n^ is reached. Nonsteady State Three Level Molecule The solutions for a three level atom under nonsteady state conditions 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,

PAGE 81

-74^=-(^3lPl3(t)^^^^31 ^^32)"3^ (B^3P^3(t) + k^2)"i + '<23"2 (III. 74) and dT = -([ ^20, li " ^21 ' ^23)^2 " ^2^3 ' ^12"l (^^^-75) It will be assumed that thermal population of levels 2 and 3 is negligible at room temperature or lower, making k-.^ = k,_ = 0. It will also be assumed that intersystem crossing from level 2 to level 3 is negligible, making k^o = 0. The following terms are defined to simplify Eqs. 1 1 1. 74 and III. 75. /\ a^ = T-^ — + k„, + k,5 (III. 76) "* ^3j,10 •^' ^^ '2[^20,li "^21 (III-77) b3^ B3^p^3(t) +B^3P^3(t) (III. 78) B = B^3P^3(t) (III. 79) Using D to denote the differential operator, Eqs. 1 1 1. 74 and 1 1 1. 75 may be written as 1 1 1. 80 and 1 1 1. 81, respectively, after substituting "1 = "t "2 "3(D + b3 + a3)n3 + Bn^ = Bn^ (III. 80) D + ap -"3"n^^"2 ° nii.81)

PAGE 82

75Eliminating the n_ term from Eq. III. 80 by multiplying (D + b + a^) times Eq. 1 11.81 and adding the result to Eq. 111,80 gives (D + b +a2)(D + a2)n2 + Bk-„n2 = Bk^p (III. 82) The solution to the homogeneous differential equation of the form of Eq. III. 82 for Pio " P for < t < t is nplt) = C^expC-a-t) + Cpexp(-a-t) + C where X

PAGE 83

•76n3(t) = n33. <^2^^2~^2^ M exp(-a„t) + a^VX -4Y exp(-a^t) + 1 (III. 90) where n„ is given by Eq. 1 1 1. 44 and n^ is given by •^SZ^Sss '2ss I '^ZOJi .+ k 21 (III. 91) At low source irradiance, a ^ a and a ^ a-, where a^ is the reciprocal of the level 2 lifetime (phosphorescence) and a^ is the reciprocal of the level 3 lifetime (fluorescence) which is the conventional low irradiance case (40) . In order to better understand the expressions for n„(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 ('x^lO" ) and a large phosphorescence quantum efficiency (^.9). Fluorene represents the case of a molecule with a moderate fluorescence quantum efficiency ('U),45) and a moderate phosphorescence quantum efficiency (^0.36). Rhodamine 6G represents the case of a high fluorescence quantum efficiency ('^l) and a small phosphorescence quantum efficiency ('^10 ). Results of calculations of log(n2/n^) and log(n-/nj) 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 n„/n, approaches the steady state value of n^/n, more slowly after n-/n^ 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

PAGE 84

77is less than or equal to the steady state saturation irradiance, the value of n»/n, increases until it reaches a value predicted by the 3level steady state model. If the source irradiance exceeds the steady state saturation irradiance, the value of n./n^ will also exceed the 3-level steady state saturation val ue of n3/nj until np/n^ 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 5 2-level model predicts a saturation irradiance approximately 10 times higher than the 3-level model for rhodamine 6G, and it is observed in Figure 6 that at 10 E , the concentration of level 3 is close to saturation. For benzophenone and fluorene, the 2-level saturation irradiance is greater tlian 10 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 ('^1 ys 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 betv;een 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. III. 89 and III. 90, the coefficients of the exponential terms may be discussed. The factor -ct^//x -4Y in Eq. III. 89 is approximately -1 and the factor a^/^X 4Y is approximately the ratio of the fluorescence rise time to the phosphorescence rise time. As the source irradiance increases above the

PAGE 85

3

PAGE 86

-79-(V^N)90n' — (^N/^N)90n

PAGE 87

s(U E ^ (U (J > (U •ICQ E O) CO O C (O SO) <4iI— O C C I— CO -ii: Ll_ C «3 4-) o o CM CO 5XJ c c: ro (O E CM •— CU CM (ji E .« ct — I — •" incM " > OJi — I — I — 111 — I CM Qj cs-iiii/ii/ji ooEcn _l -i-rooocoi/l Oi — -^ r(T) I— O 3 MCD oor^ f^ r^ I I 1 — " O OICOOOOOX CM r— n30i — 1 — I — r — 1 — XO ro S_ OI -rO r— J^ O S-+JXXXXX • ••>(£) •rU ea 1— E • X •• > c.— "=j-cMLnoo ccM 1— 03 -I3 II <0 CO .c OLncM«*CM«^ oil •<: QJ LU 1 — OLO 00 DQ rtJ II II II II II 1 — CM to SM(_) •> XJ II O 1 — O I — 1 — CMr — I — T-jllfO 14(O CnrOOOCMCMOO P>1/1 O dJ O) (tJ to «0 H> =3 > m (U =) en

PAGE 88

81 O L. O d X I I I I oa CO UJ p o CO o 00 I CVJ CO I ~iHi/^m 901' — turm 901

PAGE 89

c/) 0) rs C i o o TO fO CM 00 QJ > QJ I ^o so > -C OJ r— O ICO O OJ E rO) fO CM c so 1c QJ > (/I <0 o SQJ c c QJ U3 u c O) &so c at M OJ -a c: CM CM ' I CO 'o 3: X o vf I— Oi— rrci II II II II II 1 — LD to U •> J3 II r— I — CM 1 — I — -i-oll (13 c roncocMCMCo pco o QJ to 3 CM CO CO X » II C CM 1 — O CO o to OJ rj (O to QJ S13 en

PAGE 90

83(^N/^N)90n' — {'w^Nioon

PAGE 91

-84saturation irradiance, phosphorescence rise time decreases. When the n, concentration reaches steady state, the rise time ratio term contribution approaches zero as it is multiplied by exp(-a-,t). The terms a^ia^ a^)/{ajx^ 4Y) and a^la^ a^)/{ajy.^ 4Y) in Eq. III. 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/E . As time increases, the term exp(-aot) decreases the absolute magnitude of the negative term and the concentration of nincreases to the value allowed by the positive coefficient of exp(-apt). As exp(-apt) 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. III. 89 for n^ in Eq. III. 47 and substituting Eq. III. 90 for n^ 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 the term C is equal to unity; (ii) for low source irradiances, the luminescence radiance depends directly upon the source irradiance and the quantum efficiency;

PAGE 92

-85(iii) for high source irradiances, the luminescence radiance is independent of the source irradiance and the quantum efficiency; (iv) for all cases, the fluorescence radiance depends directly upon the total concentration of analyte, n-p; (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, n-j., can be determined by absolute measurement of the steady state Bp-value, by knowledge of kr,^ -, . , g, , g^, and by measurement of the cell path length in the direction of the detector; 3 -3 (vii) the product v .^ 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 ^-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 Af/f where f is the oscillator strength of electronic transition and Af is the oscillator strength portion attributed to the excitation transition);

PAGE 93

-86(ix) the saturation irradiance, E^, for a 3-1evel molecule at room 5 7 temperature is 10 to 10 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 2E /-nSv-,^^^^ V 1 a bci and for broad band excitation solutions, the requirement for saturation is that E of the laser source must exceed E , V V the saturation spectral irradiance equal to 2E /tt^v . ; (x) assuming saturation is reached, direct excitation of the triplet 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 ym to achieve 2 '^'MW/cm , can be used to excite many molecules with no need for 2 doubling; if >MW/cm can not be achieved and if the phosphorescence 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 n(singlet) may be exceeded under pulsed excitation conditions. The optimum measurement

PAGE 94

-87system 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 c (5t /t )E where t is the phosphorescence lifetime, t is the pulse width, and E is the saturation irradiance; this term is obtained from 1 exp(t /x ) = t /x for t /x << 1 and the factor of 5 from the fact approximately five lifetimes (risetimes) are required to reach steady state.

PAGE 95

CHAPTER IV PULSED LASER TIME RESOLVED PHOSPHORIMETRY Introduction Time resolved phosphorimetry v^as first demonstrated as a means of chemical analysis by Keirs et al . (47). They resolved a mixture of acetophenone (x = 0.008 s) and benzophenone (x = 0.006 s) at concentrations in the range of 10 to 10 M. 0' Haver and Winefordner (48) discussed the influence of phosphoroscope design on detected phosphorescence signals. St. John and Winefordner (49) used a rotating can phosphoroscope system to determine simultaneously two component mixtures. 0' 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, t., and repetition rate, f, of a pulsed source-gated detector along with the spectral shift toward the ultraviolet (52) when using pulsed xenon flashlamps should make such a system optimal for phosphorescence spectrometry. Fisher and Winefordner (53) constructed a pulsed source time resolved phosphorimeter and demonstrated the analysis of mixtures via time resolution. This system was modified to use a higher power xenon 88-

PAGE 96

-89flashlamp 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 continuous 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 phosphorimetry 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.

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-90A 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/fT )], in the denominator decreases as fx becomes greater than unity. If all else is constant and x = 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

PAGE 98

-91fluorescence 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 electronic 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 drugs and compares the use of two different lasers (pulsed nitroqen 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 (82) 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 investigators (85,86) have reported on the analytical utility of sodium iodide

PAGE 99

-92in 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 phosphorescence of 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 phosphorescence of nucleosides and found silver ion to improve detection limits 20 to 50-fold. Other metal ions (Cd(II), Hg(II), 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 1-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. Transitions 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

PAGE 100

-93of 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. 4 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 tripletsinglet transition in molecule M may mix with an "atomic like" transition in the heavy atom containing perturber, P (102); (iii) the triplet-singlet transition in molecule M mixes more strongly with an allowed transition in molecule M 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

PAGE 101

-94evidence in favor of the charge-transfer mechanism in the case of planar aromatic molecules perturbed by heavy atom containing aromatics such as tetrabromobenzene (104,105). This study reports phosphorescence lifetimes and spectra of carbazole, phenanthrene, quinine, 7,8-benzoflavone, and thiopropazate at 77 K in 10% v/v ethanol/water with different concentrations of iodide, silver, and thallous ions. Also reported is the effect of these heavy atom perturbers on the detection limits of several drugs. Experimental Instrumentation A block diagram of the equipment used in the pulsed laser time resolved phosphorimeter is shown in Figure 7. Table VII lists the model numbers and manufacturers of this equipment. The two lasers used for excitation sources were the Avco nitrogen laser and the Chromatix CMX-4 flashlamp pumped dye laser. The nitrogen laser is a super-radiant discharge giving a peak output power of 100 kW, and a pulse width measured to be 7.7 ns at 337.1 nm with an output spectral bandwidth of less than 0.1 nm. This laser was operated at 17 kV with an operating gas pressure of 17.2 torr of prepurified nitrogen in the laser channel. When a high voltage, high current pulse is rapidly applied to channel, which consists of two electrodes running the length of the channel separated by dielectric sidewalls, a transient population inversion is created by electron impact. Laser operation is achieved for a time on the order of the radiative lifetime of the upper level of the lasing transition. The laser will then radiate from both

PAGE 102

03 o
PAGE 103

96?" < X

PAGE 104

-97Table VII. Experimental Equipment and Manufacturers Item Model Number (Description) Source Laser Nitrogen Flashlamp Sample Housing Monochromator Photomultiplier Tube C950 CMX-4 H-10 4837 Photomultiplier Housing 180 High Voltage Power Supply EU-42A Gated Amplifier

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-98ends of the channel. Maximum output power is obtained using a plane mirror on one end of the channel. The divergence angle is given by the output dimension of the channel divided by tv/ice the channel length. The output beam is two lines occurring close to each electrode wall of the channel, approximately 2.5 in wide separated by 1/8 in. The flashlamp pumped dye laser was operated at 270 nm by frequencydoubling the output of laser dye coumarin 504 at 540 nm. A linear flashlamp in an elliptical cavity is used to pump the laser dye. A dye laser is a four level system. The dye is optically pumped to a higher vibrational level in the excited singlet from which it \jery rapidly relaxes to the lowest vibrational level in the first excited singlet. A population inversion relative to a higher vibrational level in the ground state results and laser action occurs. A tuning element, in this case a birefringent filter, is used to select the wavelength at which laser action occurs. Second harmonic generation, or frequency-doubling, is accomplished using a nonlinear crystal within the laser cavity to make use of the high circulating fundamental power within the laser cavity. The peak power at 270 nm is listed as 200 W with a pulse width of 1 pS. Repetition rate was restricted to 15 Hz after the destruction of a SCR in the lamp trigger circuit. As previously mentioned, the nitrogen laser output consists of two lines. It was found that the two lines would focus together into one line quite well, but along the 2.5 in length would only focus to about 5 mm. This is reasonable because the divergence along this dimension is approximately fifteen times greater than the divergence in the perpendicular direction due to the dimensions of the laser channel. The sample width is 2 mm and the long dimension of the beam is

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-99perpendicular to the sample. In order to transfer the maximum laser irradiance to the sample, the beam was rotated 90° with two front surface mirrors shown in Figure 8. The two mirrors were mounted on beam steering units which allow two coplanar rotations. The two units were mounted on a 0.5 in steel rod giving a common vertical rotation axis. The entire assembly is held in position on a steel plate with a magnetic mount. The output beam from the CMX-4 is 3 mm diameter with a divergence of less than 1 mr, so although no special optical train is required, the beam steering mirrors were used for convenience to place the CMX-4 at the same location as the nitrogen laser. The laser beam was focused on the sample by a 3 in diameter 8 in focal length lens. With the CMX-4 laser, a Corning CS7-54 filter was used to transmit the 270 nm second harmonic, but block the 540 nm fundamental. The transmittance of the output mirror used for frequency-doubling is only 1% at the fundamental wavelength. This is still a substantial amount of light, and will cause a significant interference for phosphorescence in the 540 nm range if not attenuated by a filter. When using the nitrogen laser, an interference filter with a peak transmittance of 42% at 340 nm and a bandwidth of 10 nm was used to block the nonlasing nitrogen emission lines at wavelengths greater than 360 nm. Figure 9 shows a scan of the output of the nitrogen laser from 340 to 401 nm. The beam was reflected off of an aluminum block positioned to give just full-scale at 340 nm. Figure 10 shows a scan of the output of the nitrogen laser from 360 to 540 nm using a Corning CS3-75 sharp cut yellow glass filter to block the 337.1 nm lasing line. The sample compartment is constructed from aluminum painted optical-flat black and consists of two sections. The lower section has

PAGE 107

i~ o +j ra +j o oc E IT3 (U CO O) U re %. zs 00 o s00 CT)

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-101 / / / / A

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Figure 9. Spectrum of Nitrogen Laser Output from 340 to 401 nm

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-103>" (f) liJ > 11 III J UJ S40 i60 380 400 420 WVWELENGTH (nm)

PAGE 111

c CD in E

PAGE 112

105£ LU -J Ui A1ISN31NI 3AliVn3y

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106a circular opening at the top to accomodate a Dewar flask with a quartz optical tail. The Dewar flask is held in place with a Teflon ring slipfit to the compartment opening. Nitrogen gas is flushed through the lower section to prevent condensation on the optical tail of the Dewar. A cylindrical cover fits over the Dewar and slip-fits to the lower portion of the sample compartment. On top of this cylinder an nmr spinner (84) is mounted to position the sample cell. A sample cell is a 30 cm length of synthetic fused quartz (Thermal American Fused Quartz Co., Montville, N.J. 07045) of 2 mm inner diameter and 4 mm outer diameter. The cell is fit into a Teflon cylinder using Teflon tape. The Teflon cylinder slipfits into the nmr spinner assembly. A 15 mm diameter 25 mm focal length quartz lens is used to form an image of the sample cell on the entrance slit of the monochromator. The monochromator is a f/3.5 0.1 m focal length equipped with a concave holographic grating having reciprocal linear dispersion of 8 nm/mm. Phosphorescence selected by the monochromator is detected by a photomultiplier tube in a light-tight compartment. Conventional wiring designs providing voltage to the dynodes cannot supply sufficient current to maintain linear response for the large pulses encountered using pulsed sources. They also lack the ability to keep from distorting very short (<100 ns) pulses. Lytle (113,114) has discussed photomultiplier base wiring designs to obtain fast response from photomultiplier tubes. For phosphorescence signals, the response time is easily adequate to avoid distortion, but one would like to insure that the response time is such that fluorescence or stray light does not distort the response characteristics for a long time after the light pulse has terminated. Using a linear chain of 100 kn resistors, as was originally supplied by

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-107the manufacturer, with a 1 kV supply voltage, the maximum current that can be drawn is 10 uA (assuming drawing 1% of the current flowing through the resistors does not result in nonlinearity) . Using the scheme shown in Figure 11, charge can be supplied on a transient basis from the capacitors to keep the interdynode voltage from dropping. Drawing 1% of the charge in the final capacitor for a 1 ms phosphorescence lifetime allows a peak current of 81 yA. For shorter lifetimes, the permitted peak current is higher, while for longer lifetimes the peak current is lower. It is still necessary to keep the average current 1% or less of the current flowing through the dynode resistors. Both the signal averager and boxcar integrator used require a voltage input. The input impedance of the boxcar integrator is 10 k9. and that of the signal averager is 100 kfi. Assuming a 1% loading error, the -5 minimum current measureable at the boxcar would be 2 x 10 A, and the -4 minimum peak current required to measure a lifetime would be 5 x 10 A. A block diagram of the gated current-to-voltage (I-V) amplifier is shown in Figure 12 and a complete schematic diagram in Figure 13. In order to avoid driving the front end amplifier to saturation on stray light or fluorescence, from which recovery was slow, the front end was gated using a 4016 quad analog switch. One switch is in series with the input and a second shunts the input while the other two are not used. Control voltage to the switches comes from a 4047 monostable multivibrator with complementary outputs. When the monostable is triggered, the outputs change state for a time variable from 80 to 500 ps. The shunt switch then conducts and gives a 300 n path to ground while the series switch is turned off which isolates the amplifier input by >1 Gn. When the monostable outputs return to their original state, the two switches

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Figure 11. Wiring Diagram of Photomultiplier Dynode Chain R = 100 k C = 0.002 All resistance values in ohms. All capacitance values in microfarads.

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-109-HVo

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Q. E a (U +j oi D) fO I o I +-> c (U o
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Illixi Or: UJ CL 3 9oa U -^ (.0 0-. > ^ o-> i'-i H -.1 < :j

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•113<1

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114reverse condition and essentially all the current is allowed to reach the input of the first amplifier. The 4016 was found to give smaller switching transients than any of the FET's or MOSFET's tested in this circuit. To ensure that the excitation source did not trigger while the switches were changing state, a 555 timer was triggered from the monostable output and set for a 40 ps delay before triggering a second 555 timer. The 3 ps output pulse from the second 555 timer was used to trigger the excitation source. The first amplifier is a 40J which gives a voltage output equal to minus the input current times the feedback resistance. The second amplifier is a 741 with variable gain and is clamped to respond only to positive input voltages. The output of the variable gain amplifier is output through a buffer amplifier and through an active low pass filter with time constants varying from 0.03 ms to 340 ms. The output of the active filter is offset by -3.00 V for use with the signal averager. The boxcar integrator input was connected to the unfiltered output. If a voltage proportional to the excitation source output is available, an analog divider may be switched in to ratio the signal to the source intensity. Such a system will compensate for source output variation. A Wavetek signal generator was used to trigger the monostable in the amplifier and control the laser repetition rate when using the nitrogen laser. The CMX-4 is externally triggerable only in phase with line frequency. A 10 V square wave output in phase with line frequency is provided from the CMX-4. This output was level shifted with a 4050 buffer and divided down to the desired frequency using one or two 4018 4 N counters depending on the desired frequency. This signal was used to trigger the amplifier monostable.

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-115For lifetime measurements, the filtered amplifier output was connected to the signal averager. This signal averager has sweep times variable in factors of 2 from 5 ms to 81.92 s and delay before the start of the sweep in factors of 2 from 0.32 ms to 5.12 s. The delay before may also be zero. The input analogto-digital converter (A/D) operates in the range of -3.15 to 3.20 V. The conversion factor is one count per 50 mV. The -3.00 offset on the amplifier allowed maximum use of the available A/D dynamic range. The signal averager acquires 1000 points during each sweep and sums the value acquired at each point into memory. To select the proper repetition rate, sweep time, and amplifier gain, the signal was first displayed on an oscilloscope. After the desired number of sweeps have been averaged, the contents of the signal averager memory are output to the paper tape punch in the form of 16 bit words. Paper tapes were read into a POP 11/20 minicomputer at a later time for calculation of lifetimes. For quantitative analysis, a PAR CW-1 boxcar integrator is used. Despite the term integrator in the name, a more appropriate name would be boxcar averager because the output is proportional to the average value of the input signal during the gate width. The boxcar and signal averager are both triggered from the same pulse used to trigger the laser. The delay before and gate width are continuously variable from 1 us to 1 s. The output from the boxcar integrator is displayed on a strip chart recorder. Instrumental Procedure For all intensity measurements and lifetime measurements, the monochromator slit width was 2.0mm (giving a spectral bandwidth of 16 nm)

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-116For scanning the phosphorescence spectra, the slit width was decreased to 0.5 mm for a spectral bandwidth of 4 nm. All spectra are uncorrected for variation in response with wavelength. The photomultiplier voltage was -800 V and the amplifier was gated off for 200 us in all experiments. In all lifetime determinations, the sweep time was set to four or five times the lifetime estimated from the oscilloscope trace. For lifetimes of 1 s or less, 128, 256, or 512 sweeps were averaged. For lifetimes longer than 1 s, the nitrogen laser was run at 20 Hz (CMX-4 at 15 Hz) until the phosphorescence signal reached steady state; the laser trigger was turned off and the phosphorescence decay for one sweep was stored in the signal averager. For determining the short component in a two component or nonexponential decay, the sweep time was set long enough to allow the short component to decay completely and establish a baseline of the long component. To determine the long component, a delay before the sweep time was set long enough to allow the short component to decay completely. Although a long enough delay to allow an intense short component to decay completely is possible, the amplifier still has a finite dynamic range. Recovery time for amplifiers driven to saturation is dependent on how long the amplifier has been saturated. Increasing the time constant will prevent saturation and will result in the short component decay having a lifetime equal to the time constant (assuming a time constant greater than the short component lifetime). This in turn increases the required delay time. These effects make the measurement of a long lifetime less precise if an intense short component is present.

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117For studies of the external heavy atom effect, the nitrogen laser was operated at 20 Hz, the boxcar integrator delay time set to 0.2 ms, and the gate width to 1 ms. This was done to maintain a roughly constant detection duty cycle as the lifetimes changed. The observed time constant (OTC) of the boxcar integrator is given by Observed Time Constant = ^ , u-!!!!.^°"'n^"^ .• fr^ Gate Width x Repetition Rate With a 20 Hz repetition gate and 1 ms gate width, a 30 ms time constant to give an OTC of 1.5 s was used. For limits of detection, the delay time and gate width were varied to maximize the S/N ratio, and the OTC set as close to 3 s as possible. Data Reduction Fisher (115) and Harbaugh (60) have described methods of data reduction for single and mul ticomponent decays. The basic theme of all involves reading decays off of a chart recorder and either plotting the data on semi-log paper or taking the natural log of the relative intensity and using a linear least squares to fit the decay curve. It is impractical to try and read 1000 points from a chart recorder tracing of the decay curve, so most of the data is wasted. Digital data are already available from the signal averager, so the effort expended in re-digitizing the data by eye is not justified. To utilize the maximum amount of data available in the most reasonable amount of time, programs were written in BASIC for a PDP 11/20 minicomputer. Data are read into the computer from paper tape and stored on floppy disks in a virtual file as 16 bit binary words. Lifetimes

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118were calculated from the least squares slope of the natural log of the phosphorescence signal versus time after subtracting the blank. The short component of a two component decay was calculated after extrapolating back the long component using a previously calculated lifetime of the long component. Programs used are listed in the Appendix. Reagents All chemicals used in this study were reagent-grade and used as received. Controlled substances were ordered and used in accordance with BNDD regulations. Chemicals and drugs used in this study were obtained from the following sources: carbazole, K & K Laboratories Inc., Plainview, N.Y.; 7,8-benzoflavone, phenanthrene, Eastman Kodak Co., Rochester, N.Y.; benzophenone, sodium iodide, Fisher Scientific Co., Fairlawn, N.J.; silver nitrate, potassium iodide, Mallinckrodt Chemical Co., St. Louis, Mo. ; thiopropazate, Searle Co., San Juan, Puerto Rico; phencyclidine, Phillips Roxane, Inc., St. Joseph, Mo.; 2,5-dimethoxy-4-methylamphetamine (STP or DOM), NIMH, Center for Studies of Narcotics and Drug Abuse, Rockville, Md.; vinblastine sulfate, Eli Lilly and Co., Indianapolis, In.; phenylbutazone (butazol idin) , sulfinpyrazone (anturane), oxyphenbutazone (tandearil), Ciba-Beigy, Summit, N.J.; quinine, morphine, ethyl morphine, codeine, procaine, phenobarbitol , cocaine, Applied Science Laboratories, State College, Pa.; ethyl alcohol, U.S. Industrial Chemicals Co., New York, N.Y.; deionized water from a Barnstead Nanopure water system, Barnstead, Boston, Mass. Structures of drugs have been previously given (60,94).

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119Stock solutions of phosphors were prepared in ethanol and stock solutions of sodium iodide, potassium iodide, and silver nitrate in water. Solutions for analysis were prepared by diluting 1 ml of the ethanol stock solution and the appropriate volume of other reagent solution and filling to the mark in a 10 ml volumetric flask. In all studies, a 10/90 v/v ethanol/water solvent was used. The concentration of heavy atom perturber given is that at room temperature in the mixed ethanolwater solvent. Results and Discussion External Heavy Atom Effect of Iodide, Silver, and Thallous Ions Silver nitrate (18,93), sodium iodide and potassium iodide (85,88), and thallous nitrate (19,116) were chosen for investigation as heavy atom perturbers based on previous reports of analytical utility. No difference was found between sodium and potassium iodide with respect to the heavy atom effect, but potassium iodide was found to give lower phosphorescence background. Carbazole (93), phenanthrene (94,112), quinine, 7,8-benzoflavone, and thiopropazate were chosen for study from previous reports of interaction with heavy atom perturbers and to represent several different classes of molecules. Lifetimes, correlation coefficients, intensity ratios for the long and short components of the phosphorescence, and the ratios of the phosphorescence signal with and without heavy atom perturbers are given for the molecules studied in Tables VIII through XII. The ratio of shorter 1 ived component to the longer component is dependent on the lifetimes, laser pulse width, and repetition rate of the laser when using the signal averager. The

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-120excitation duty cycle, d , is given by (see Table VI, time resolution C A section) t ^P, . P P ^ex % Li expf-l/fx )] ^^^-^^ assuming t << t (10 ns << 1 ms), where x is the phosphorescence life^ p p p '^ '^ time. As the phosphorescence lifetime decreases, this term increases. For f = 20 Hz and t = 10"^ s, d increases from 2 x lO"^ to 1 x lO"^ p ex as T decreases from 1 s to 1 ms. The ratio Ir/h is defined as the ratio of the initial intensity of the short component to the initial intensity of the long component at the listed repetition rate. Also, I,-/!, is defined as the ratio of the initial intensity of the short component to the initial intensity of the long component after dividing each by its respective d . In equation form, this is d (short)l! I d (long)!-r ("-2) ex^ ^ L L The ratio lo/Ij^ would be observed for continuous wave excitation. The ratio I/I is defined as the ratio of the phosphorescence signal with heavy atom perturber to the phosphorescence signal without heavy atom perturber where both signals have been measured using the boxcar integrator with f = 20 Hz, t =1.0 ms, and t. = 0.2 ms. This g d reflects the increase (decrease) in phosphorescence signal using the heavy atom effect in pulsed laser time resolved phosphorimetry. The portion of the duty cycle (Table VI, time resolution case) associated with gated detection, d ,, is given by Tp[1 exp(-t 7T)]exp(-tVT) 'gd ' t. d.. =-B :-3 d (jy_3j

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•121 Under these conditions (f = 20 Hz, t =1.0 ms, and t, = 0.2 ms), if the lifetime changes from 10 s to 1 ms, d , changes from 1.0 to 0.52. Because the gated detector averages over the gate time, using a gate time much shorter than the lifetime does not decrease the signal at the boxcar output. Using a gate time longer than the lifetime does decrease the signal at the boxcar output. This does not mean that the conditions used give the optimum S/N ratio, but it does eliminate the need to change the gate time unless the lifetime gets below 1 ms. The ratio I|/IP is still influenced by the excitation duty cycle. The normalized heavy atom enhancement factors listed in Table XIII are the ratios of the phosphorescence signals with and without the heavy atom perturber, after each signal has been divided by its respective excitation and detection duty cycle (or divided by d.^, time resolution case, in Table VI). This value now reflects the ratio of the phosphorescence signal with heavy atom perturber to the ohosphorescence signal without heavy atom perturber with continuous wave detection and emission. It is an approximation of the change in phosphorescence quantum efficiency, but it is only an approximation because the entire phosphorescence spectrum has not been integrated (only measured at the given wavelength with a 16 nm spectral bandwidth). Due to the nonexponential decay of phosphorescence enhanced by the external heavy atom effect, calculation of lifetimes presents a problem. McGlynn et al . (100) have reported first observable half lives, which are the half lives of the earliest portion of the decays measurable. Lifetimes of room temperature phosphorescence have also been reported in this v/ay (117). This lifetime is dependent on the delay between the termination of excitation and the first observation of phosphorescence

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-122Ol (^ >— o o ^ M en n3 cC ST3 ra C to -t-J a a .— ^ OJ LO c E o V — a. E OJ o E O +-> +-> OJ sMO •I-C _l U~l (X3 -M C OJ — o Ol E E O •IO O) CD MC •IO o s+-> c U c o ^ 00 CO • . • o I— CO ooi — cMOCM^LnLD <^ CO C\J (NJ CM I I I I I o o o o o X X X X X «;JVO en LT) I — cTi I— n r^ o o r-~ -^ <£) «;}o o o o o o o o CO CM O O I — r— X X CM LO I I— r^ CO CO r— I— I— CO CO r^ o o o o o ts4 o CM I 1^ "=1* o I r~00 1^ CO o o o o cri LD 00 CO CO r-^ LO ^100 CO t~-> r~-un I o cTi en en en cTi CTi cr> I I o o o o o o o r — (^ r-^ CO o r^ r^ CM r— r— I— CM 1— r—

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126CO o (t3 o o Ln (U c > -M (D Ol oi -o c 01 3. 1^ CM o N c Ol CQ I 00 X 0) .a u o . 00 +-) ^— ~ dJ E O -— ' QE O) o _E o 4-> +-> (U sMO •rx: _l 10 +-> 3 c ,-^ q; 1/1 C O Q OJ CD 4C •-O o (t3 S+-> c u c: o (_) 1— «ao o r^ r— I — CO LO ^ 0000 N 00 ir I CO .— cTi 00 c\j I 00 o 1 — 1 — CM r-~ CTl CVJ CTl i-D r^ to «^ LT) UD Ln CTl CTl CTl 00000 CTl >=1.— CM 1— r~~ r~r-«. «D vo r-^ c\j Ln en en CT) CTi 00 Ol CTi CTv CT> CTl CTi CT) • • • o o o o o o^— — "^ 00
PAGE 134

127O en c to 00 ai +-> > +-) CD ex. (O 4J 3. un +J to M n3 QO i. CL o o o . o 00 D C — > o Eo "4o •rx: 3

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•128Table XIII. Normalized Heavy Atom Enhancement Factors Enhancement Factor Carbazole Phenanthrene Quinine 7,8-Benzoflavone Thiopropazate 0.8 M Nal 0.01 M AgNO^ 0.8 M Nal 0.1 M AgNO^ 0.7 M Nal 0.1 M AgNO^ 0.3 M KI 0.1 M AgNO^ 0.75 M KI 0.1 M AgNO^ 9. 5.

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-129which win vary for different experimental systems. Najbar and Chadhowska (111) have observed for triphenylene in the presence of KI in an ethanol glass at 77 K that the lifetime becomes exponential with the same lifetime as unperturbed triphenylene after approximately two lifetimes. The long component reported here is the longest lifetime measurable subject to the dynamic range considerations of the amplifier and signal averager. Short component lifetimes were all calculated after subtraction of the long-lived component. Results for individual molecules will be discussed first. Carbazole shows a substantial reduction in phosphorescence lifetime for Nal and AgNO^ perturbers, but little effect for TINO^. The long component lifetimes with added Nal are reduced, and the correlation coefficients are as good as those for unperturbed carbazole. The shortlived component is certainly nonexponential . Using silver nitrate as the perturber gives the opposite case. The short component appearing at 1.0 X 10" M grows in intensity until 1.0 x 10"^ M AgNO^ after which it begins to decrease. The lifetime of the long-lived component shortens much less than with Nal while the intensity is much smaller than the short-lived component. The poorer correlation coefficients of the long lived component may reflect nonexponential character, but it is more likely that they reflect the difficulty in measuring a longlived species in the presence of a much more intense shorter component. The normalized enhancement of phosphorescence of 9.6 for 0.8 M Nal and 5.0 for 0.01 M AgNO^ reflects a difference between the two heavy atom perturbers. The intersystem crossing efficiency (Eq. III. 34) for carbazole is reported as 0.36 (118); so regardless of how large the intersystem crossing rate constant (k^2 ^^ '^TSf ^~ ^ becomes, the

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-130phosphorescence quantum efficiency can increase at most by a factor of 2.8 due to intersystem crossing. The phosphorescence quantum efficiency (Eq. III. 33) may also increase if the phosphorescence transition probability, A„, , increases relative to the radiationless rate constant, kp, . The phosphorescence lifetime is (A„, + kp-,)" . From the ratio of the lifetimes in 0.75 M KI (0.23 s) and without KI (6.8 s), the sum (A„, + kp,) has increased a factor of 30 in 0.75 M KI. Because there is more increase in the normalized enhancement than can be accounted for by increase in intersystem crossing efficiency, A„, has increased more than kpi . Studies of the internal heavy atom effect and the external heavy atom effect have indicated (100) that Ap, is affected more than k^, . In 0.01 M AgNO^, the phosphorescence lifetime has decreased more than a factor of 3000, but the normalized enhancement factor is only 5.0 which is less than that found for iodide. If it is true that kr^-p is the parameter most sensitive to the external heavy atom effect (100), then silver ion must have a greater affect on k , than does iodide or the normalized enhancement would be the same. Silver ion certainly affects the sum (A„, + kp-i ) to a greater extent than does iodide ion. The decrease in phosphorescence signals for silver nitrate concentrations above 0.01 M could be due to an increase in k„, , but the signal levels are approaching the point of nonlinear photomultiplier response. The phosphorescence spectra of carbazole, carbazole in 0.75 M KI, and carbazole in 0.1 M AgNO^ are shown in Figure 14. All of the bands resolvable without Nal appear to be present in 0.75 M Nal with the intensities of the bands at 422, 438, and 453 nm increased in intensity relative to the 409 nm band with the band at 422 nm enhanced to the

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-131largest degree. In 0.1 M AgNO^, only two broad bands centered at 426 and 447 nm are resolved. The interaction of phenanthrene with AgNO^ and Nal perturbers is yery similar to that of carbazole. The interaction of phenanthrene with KI in ethanol glass at 77 K (112), fluorescence quenching at room temperature by KI in acetonitrile (119) and in bromobenzene (120), and phenanthrene phosphorescence with ethyl iodide and iodonaphthalene as heavy atom perturbers have all been reported. The lifetimes measured for phenanthrene perturbed by Nal are qualitatively in good agreement with the results of Najbar et al . (112). Lifetimes of the long component reported here are somewhat shorter than theirs for the same iodide concentration. The results in a snovyed-matrix which is mainly aqueous are not likely to be directly transferrable to results in an ethanol glass for the same iodide concentration. The concentration of phenanthrene studied by Najbar et al. (70) is not stated, but results for naphthalene are given at 10 M, or 100-fold higher than the concentrations studied here. Becker (44) tabulates the intersystem crossing efficiency for phenanthrene as 0.88 and 0.76 and the phosphorescence quantum efficiency as 0.20, 0.11, and 0.09 depending on how the measurements were made. Little increase in the intersystem crossing efficiency is possible, so the majority of the increase in phosphorescence efficiency must come from an increase in the phosphorescence transition probability. The normalized enhancement factor of 5.1 in 0.8 M KI corresponds to a 6-fold decrease in lifetime, so radiationless deactivation of the triplet seems enhanced only to a small extent. Results with AgNO^ perturbing phenanthrene are also similar to carbazole. The normalized enhancement factor is smaller probably due to the already high intersystem crossing efficiency in phenanthrene.

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-132Phosphorescence spectra of phenanthrene without heavy atom perturbers, with 0.75 M KI, and with 0.1 M AgNO^ are shown in Figure 15. Broadening of bands by silver nitrate is again observed, and no change of relative band intensities in 0.75 M KI is observed. At this point, it should be mentioned that 0.1 M TINO^ was also investigated as a heavy atom perturber on phenanthrene with results very similar to those obtained for carbazole. Equally insignificant phosphorescence enhancement was observed for quinine, at which point, investigations using T1N0-, were discontinued. The effect of iodide and silver ions on quinine phosphorescence is markedly different than the effect on carbazole or phenanthrene. Quinine fluorescence is known to be sensitive to quenching by halide ions (121). The trends in the values of the lifetimes with iodide as the heavy atom perturber are consistent with the data for phenanthrene and carbazole, but the intensities are not. The short component is observable at 100-fold lower iodide concentrations and is 100-fold more intense relative to the long component. Adams et al . (80) observed complete -3 quenching of quinine fluorescence at 10 M CI and a 1.7-fold increase in the photoacoustic signal. The fluorescence quantum efficiency measured was 0.52. It would seem that most of the fluorescence quenching goes through the quinine triplet and a 2-fold upper limit on the increase in intersystem crossing efficiency seems reasonable. Extrapolating from information obtained with 0.1 M HCl and 0.1 M H^SO, should be treated with caution. The point to be made is that it is again unlikely that all of the increase of the normalized enhancement factor can be attributed to an increase in intersystem crossing efficiency.

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-133Silver ion also has a markedly different effect on the lifetime and intensity. A gradual decrease in the lifetime of the long component is _5 not observed, but rather a sharp decrease in the lifetime at 3.3 x 10 M _2 AgNOo. A shorter component is observed starting at 3.3 x 10 M AgNO^ of lower intensity relative to the long lifetime than previously discussed cases. Quenching of quinine fluorescence at room temperature by AgNO^ was investigated using an Aminco-Bowman spectrofluorimeter. Fluorescence is quenched less rapidly than the phosphorescence is enhanced as is shown by comparison of the results in Figure 16 with the tabulated increase in phosphorescence. Nitrate ion has been observed to quench the room temperature fluorescence of aromatic hydrocarbons (122), so it is possible that some of the quenching is due to nitrate ion. The lack of change in the phosphorescence of quinine by 0.1 M TINO^ would serve to confirm that the effect on phosphorescence is due to silver ion rather than nitrate. The conclusion is that the sharp drop in phosphorescence lifetime at 3.3 x 10 M AgN0-< is not accompanied by any major increase in intersystem crossing efficiency and that the phosphorescence transition probability is more sensitive to perturbation by silver ion than the intersystem crossing rate constant. The phosphorescence spectra of quinine perturbed by Nal and AgNO^ are shown in Figure 17. The bands at 474 and 509 nm in 0.7 M Nal have the same intensity ratio as the bands at 474 and 497 nm of unperturbed quinine, while the 463 nm band in unperturbed quinine is a shoulder in the Nal perturbed spectrum. The 509 nm band may correspond to the 509 nm shoulder in unperturbed quinine, and the similar intensity ratio may be coincidence. The spectrum of quinine perturbed by AgNO^ has two bands at 477 and 509 nm which could correspond to the 474 and 509 nm

PAGE 141

-134bands or red shifted and enhanced 463 and 497 nm bands from unperturbed quinine. Results obtained for 7,8-benzoflavone (BF) are intermediate to the previously discussed cases. The lifetimes of the short component of the decay in KI perturbed BF are longer than those obtained for quinine, _2 while the majority of the intensity starting at 1.0 x 10 M KI is in the short component. The normalized enhancement factor in KI is lower, but the phosphorescence quantum efficiency has been reported as 0.45 (122). This would make the enhancement of 3.4 in 0.3 M KI seem too high. The reported quantum efficiency may be high and the factor of 3.4 is not a true quantum efficiency as the phosphorescence spectrum has not been integrated. There is no observed fluorescence for BF due to highly efficient intersystem crossing from the n,Tr* lowest singlet to the 71,77* triplet (123,124). At concentrations of KI higher than 0.3 M, solutions turned yellow due to formation of iZ. After standing for several hours, a blue-green precipitate was observed. To avoid complications, only freshly prepared solutions were used and 0.3 M was the maximum KI concentration studied. Instability of BF in solution has been reported (123). Lifetimes with silver ion as a perturber are consistent with carbazole and phenanthrene data with smaller interaction in terms of the short component lifetimes and the ratios of the short and long component intensities. The spectra of perturbed and unperturbed BF are shown in Figure 18. The 459 and 493 nm bands have been enhanced in 0.3 M KI relative to the band at 471 nm. The spectrum in 0.1 M AgNO„ appears as a broadened and slightly red shifted version of the unperturbed spectrum.

PAGE 142

135Thiopropazate perturbed by KI shov/s essentially no change in lifetime and a weak nonexponential short component. The presence of a chlorine and sulfur in thiopropazate will give an internal heavy atom effect resulting in the short lifetime of the unperturbed molecule. The normalized enhancement factor of 1.2 in 0.75 M KI is also consistent with this. The spectra of externally perturbed and unperturbed thiopropazate is shown in Figure 19. The spectra of thiopropazate with and without KI is a broad structureless band centered at 500 nm. The lifetime data for thiopropazate perturbed by AgNO., is not consistent with any of the other cases, nor is the 25 nm blue shift observed in the phosphorescence spec-4 -3 trum. The decrease for 1 x 10 and 1 x 10 M AgNO^ of the phosphorescence intensity is due to the fact that the band is shifting. An interesting application of the external heavy atom effect is to study the triplet states of molecules that fluoresce, but do not phosphoresce. At 77 K, riboflavin has intense green fluorescence, but in 0.75 MKI, riboflavin has no fluorescence and bright orange phosphorescence. The work by Azumi (104) and Yamauchi et al . (105) clearly explains the nature of the heavy atom effect assuming a planar molecule and an external heavy atom perturber. Treating the three triplet spin subcomponents separately, the x, y, and z subcomponents transform as rotations about the x, y, and z axes. One-center integrals mix the y and z subcomponents with a,i\* or Tr,o* singlets and three-center integrals mix the X subcomponent, so the y and z subcomponents should contribute the majority of the intensity found in the triplet. For the moleculeperturber pair, a charge transfer state from a tt orbital of the molecule to a o* orbital of the perturber will mix two-center terms with the x, y, and z subcomponents. The new two-center terms are larger than the

PAGE 143

Figure 14. Phosphorescence Spectra of Carbazole Carbazole concentration 0.95 yg/ml Carbazole in 10/90 v/v E/W f = 20 Hz, t , = 9 ms, t = 10 ms, 10 ' A Full Scale. d g ( ) Carbazole in 10/90 v/v E/W .^ ( ) Carbazole in 10/90 v/v E/W, 0.75 M KI -, f = 20 Hz, t, = 9 ms, t = 10 ms, 2x10" A Full Scale, a g (• • •) Carbazole in 10/90 v/v E/W, 0.1 M AgNOo f 20 Hz, t. = 0.2 ms, t =1 ms, 5x10-6 A Full Scale. ' d ' g

PAGE 144

-137tr P r i O 2: LiJ O Id O o mevyxa lA 400 450 550 LENGTH (hri

PAGE 145

Figure 15. Phosphorescence Spectra of Phenanthrene Phenanthrene concentration 1.5 ug/ml . ( ) Phenanthrene in 10/90 v/v E/W f = 30 Hz, t , = 4 ms, t = 15 nis , 2 x 10"^ A Full Scale, d g ( ) Phenanthrene in 10/90 v/v E/W, 0.75 M KI f=30 Hz, t, = 4 ms, t =15 ms, 5x10-8 A Full Scale, d g (• • •) Phenanthrene in 10/90 v/v E/W, 0.1 M AgN03 f = 30 Hz, t . = 0.2 ms, t =1.0 ms, 5 x 10-7 a Full Scale, d ' g

PAGE 146

-139b UJ o UJ o UJ o X DL CO o X LLi > < -J UJ 425 575 WAVELENGTH (nm)

PAGE 147

Figure 16. Quenching of Quinine Room Temperature Fluorescence bv AgNO^ Quinine concentration, 4.3 yg/ml ; Excitation wavelength, 310 nm; Emission wavelength, 382 nm.

PAGE 148

141 if) LlI UJ O I UJ O O o o (/) UJ o O O I 1 I I I -5.0 -4.0 -3.0 -2.0 -1.0 LOG AgNO, CONCENTRATION (M)

PAGE 149

Figure 17. Phosphorescence Spectra of Quinine Quinine concentration 4.3 yg/ml . ( -) Quinine in 10/90 v/v E/W f=20 Hz, tj = 0.4 ms, t =10 ms, 5x10-8 a Full Scale. ( ) Quinine in 10/90 v/v E/W, 0.75 M KI f=20 Hz, t, = 0.2 ms, t =2.0 ms, 5x10-6 A Full Scale, d g (• • •) Quinine in 10/90 v/v E/W, 0.1 M AgNOs f = 20 Hz, t . = 0.2 ms, t = 2.0 ms, 2 x 10-6 A Full Scale, d ' g

PAGE 150

-143425 W^^ELErCTH (nm)

PAGE 151

Figure 18. Phosphorescence Spectra of 7,8-Benzoflavone BF concentration 2.7 yg/ml . ( ) BF in 10/90 v/v E/W f-20 Hz, t^-1 ms, tg = 9 ms, 10-7 A Full Scale. ( ) BF in 10/90 v/v E/W, 0.3 M KI f=20 Hz, t^ = 0.2 ms, t^ = 1 ms, 10-6 a Full Scale. (• • •) BF in 10/90 v/v E/W, 0.1 M AgN03 f=20 Hz, t^ = 0.2 ms, tg = 2.0 ms, 10-6 a Full Scale,

PAGE 152

145425 •',75

PAGE 153

Figure 19. Phosphorescence Spectra of Thiopropazate Thiopropazate concentration 5.2 yg/ml . ( ) Thiopropazate in 10/90 v/v E/W f-20 Hz, t^ = 0.2 ms, t =6.0 ms, 2x10-7 A Full Scale. ( ) Thiopropazate in 10/90 v/v E/W, 0.75 M KI f-20 Hz, t^-0.2 ms, t =4.0 ms, 2x10-7 A Full Scale. (• • •) Thiopropazate in 10/90 v/v E/W, 0.1 M AgN03 f = 30 Hz, t^ = 0.2 ms, t =2.0 ms , 1x10-6 A Full Scale.

PAGE 154

b m rs" viu«» -147U a: o if*, o iLi \V •rT 425 525 575 W/WELEMGTH (nm)

PAGE 155

-148three-center terms mixing v/ith the x subcomponent, but are not likely to be larger than the original one center terms. Experimental results for quinoxaline confirm this treatment. Extension to include vibronic coupling indicates that those vibronic bands which originally gain intensity in the triplet manifold are enhanced by an external heavy atom perturber while those vibronic bands which gain intensity by vibronic coupling in the singlet are not perturbed. In summary, the external heavy atom perturber opens new spin-orbit coupling channels and which vibronic bands are enhanced depends upon the manifold from which intensity is gained. It is difficult to visualize a charge-transfer complex with iodide ion acting as an electron acceptor, but it might be expected to act as a donor. No chargetransfer absorption bands in the room temperature spectra of napthalene and phenanthrene in the presence of KI have been observed (112). Azumi (104) proposes that the charge-transfer complex need not be stable, while Najbar et al. (112) state that a stable charge-transfer complex is a requirement. They report that in 0.125 M KI, the phenanthrene spectrum has three components. One is unperturbed phenanthrene, one is heavy atom enhanced phosphorescence, and the third is due to asymmetric CH vibrational mode. In 0.7 M Nal with a lower concentration of phenanthrene, the unperturbed component is likely to be much smaller due to the heavy atom perturbed component, and the component due to the asymmetric vibrational mode is also of lower intensity. The spectra of carbazole, quinine, and BF perturbed by iodide, along with the lifetime data are consistent with a picture of the external heavy atom effect enhancing different vibrational modes to a different extent depending on the way in which the bands gain intensity in the unperturbed molecule.

PAGE 156

149The data for thiopropazate with iodide are consistent with a small effect expected on a molecule already containing an internal heavy atom. Substitution of a chlorine on napthalene decreases the phosphorescence lifetime from 2.3 s to 0.29 s, and the lifetime of the sulfur containing analog of carbazole, dibenzothiophene, is 1.3 s (100). An unperturbed lifetime of 67 ms seems reasonable for a molecule containing both a sulfur and chlorine, but an n,iT* triplet would also be unperturbed by an external heavy atom. For example, no decrease in the lifetime of benzophenone was observed in 0.75 M KI. Najbar et al . (112) have assigned the heavy atom enhanced component of aromatic hydrocarbons to the purely electronic exchange mechanism based on observations of Giachino and Kearns (108,109) that the phosphorescence spectrum should appear unchanged and include only symmetric vibrational modes. They attribute the enhanced asymmetric vibrations to a third order perturbation involving electron exchange, spin-orbit coupling, and vibronic coupling. Azumi (104) has pointed out that for purely electronic exchange, the locally excited states of the perturber contribute to the transition moment proportional to the fourth power of the intermolecular overlap while chargetransfer states contribute terms approximately proportional to the second power of the overlap. Azumi (104) also noted that this formalism may require modification in cases where the chargetransfer configuration is not easily conceived of. In the case of silver ion, complexes with olefins and aromatic compounds are well-known (125). The crystal structure of [C^H^-Ag] CIO. has been determined, and the silver ion is located asymmetrically with respect to the ring (126). The bonding is normally considered to consist

PAGE 157

150of one component involving overlap of the it electron density of the aromatic with a o-type acceptor orbital on silver ion and a "back bond" from filled d or dTT-p?: hybrid orbitals into antibonding orbitals on the carbon atoms. The donation of -n electron density to the metal a orbital is considered the larger effect, so the interaction may be thought of in terms of an aromatic donor and metal ion acceptor chargetransfer complex. For carbazole, phenanthrene, quinine, and BF, the phosphorescence spectra in 0.1 M AgNO^ have the same components enhanced as those enhanced by iodide ion v/ith substantial broadening in the silver nitrate case. The absorption spectrum of toluene complexed with silver ion is slightly red shifted and broadened compared to the absorption spectrum of toluene (127). The anomaly of the quinine lifetimes perturbed by AgNO^ may be explained by the binding of silver ion at the ring nitrogen. This is consistent with the fact that silver ion binds at both the double bond and ring nitrogen in 2-allyl pyridine (125). Similar effects have been observed on the phosphorescence of purine nucleosides (18). The ring nitrogen binding occurs at lower concentration of silver ion than the complex with the aromatic portion of the molecule. The blue shift of the phosphorescence and behavior of the lifetimes of thiopropazate does not seem to fit in. Silver ion has been shown to interact strongly with sulfhydryl proteins (128,129) so binding of silver at the sulfide is possible. The energy level of the lowest triplet has certainly increased, so the relative positions of the singlet and triplet tt.tt* and n,7i* levels may change. A mixed 7r,ii* and n,TT* triplet may also account for the short phosphorescence lifetime of thiopropazate. Orbital inversion and vibronic coupling have been observed to be important for a number of aza aromatics and aryl ketones

PAGE 158

•151(130). A systematic study of phenothiazine and phenoxanthin along with the respective halo substituted compounds is required to resolve this question. Certain aspects of the heavy atom effect on room temperature phosphorescence can be discussed based on these studies. Iodide has been observed to be a more effective heavy atom perturber on carbazole than is silver ion (93). This is consistent with the results reported here and the fact that in strong bases (1 M NaOH) carbazole is ionic. Phenanthrene is the opposite case, with silver ion more effective than iodide. To further compare results, the room temperature phosphorescence lifetime of phenanthrene in 0.1 M AgNO, was measured and found to be 12 ms, which is longer than the lifetime at 77 K. Thallous ion has been observed to be a more effective heavy atom perturber on phenanthrene and other aromatic hydrocarbons than silver ion (19). In view of the small effect of thallium ion at 77 K it would appear that some other factors, such as the way the molecule is held on the paper at room temperature, are also of major importance. Caution should be applied in interpreting room temperature phosphorescence using thallium and silver ions strictly in terms of the heavy atom effect. The comparison of results for iodide and silver ion as heavy atom perturbers indicate that a stable charge-transfer complex with an aromatic molecule acting as the donor is not required for the external heavy atom effect to operate. The lack of effect for thallium ion relative to silver ion indicates that something besides a heavy atom in the vicinity is required. For inorganic ions, the energy difference between levels suitable for mixing is likely to be as important as the atomic number. The formalism of Azumi (104) is impressive for cases

PAGE 159

152where the charge-transfer configuration is accessible, but it can not in its present form explain the iodide results as a heavy atom perturber. The exchange mechanism does predict broadening (108,109), so it would appear to offer the best explanation for the results of iodide ion in the heavy atom effect at the present time. Silver has a lower atomic number than iodine, so a smaller heavy atom effect would be predicted. In all cases studied here, silver ion had a larger effect on the phosphorescence lifetime than iodide. This could indicate that a "chargetransfer complex" heavy atom effect is a stronger perturber than an "exchange" heavy atom effect. Lifetimes and Limits of Detection for Several Drugs The lifetimes of the drugs investigated in this study are given in Table XIV along with other pertinent information on the experimental conditions. In cases where a single exponential decay was found, the lifetime is arbitrarily placed under the Long Component heading. For several of the drugs present as hydrochlorides, it was not possible to use 0.1 M AgNO^ due to precipitation of AgCl . Morphine, ethyl morphine, and codeine all have relatively short phosphorescence lifetimes and showed weak phosphorescence. The observed increase in the lifetime of morphine in 0.75 M KI is likely an error due to the weak signal. Morphine gave the poorest signal of all the drugs studied. The lifetimes of all three were reduced in 0.1 M AgNO^ along with an increase in phosphorescence signal. All other drugs, except the phenylbutazone derivatives anturane, butazolidin, and tandearil, have decreased lifetimes and nonexponential decays in 0.75 M KI. Anturane,

PAGE 160

•153i. o O CD o -M O c la .c 4-> UJ > > O cr> c

PAGE 161

154O) :3 c M C o u X

PAGE 162

-155like morphine, gave an increased lifetime in 0.75 M KI. The reason, in this case, is most likely due to a photochemical reaction. Under laser excitation, most other compounds gave slowly decreasing signals with time, but the phosphorescence of anturane increases with time. The rate of increase qualitatively appeared to be greater in 0.75 M KI. Butazolidin phosphorescence decreased in 0.75 M KI, and tandearil solutions rapidly turned yellow from tri iodide formation. Phencyclidine and cocaine showed the largest increase in phosphorescence signal and the greatest decrease in lifetime. The "short" component of cocaine's phosphorescence decay having a 0.22 ms lifetime was the most intense of all the "short" components of nonexponential decays. Vinblastine sulfate is an antitumor alkaloid isolated from Vinca Rosea . It is the only drug studied which showed any vibrational structure in the phosphorescence spectrum, shown in Figure 20. Also shown in Figure 20 is the phosphorescence spectrum of vinblastine sulfate in 0.1 M AgNO-. The broadening of the phosphorescence spectrum and sharp decrease in lifetime in 0.1 M AgNO^ are consistent with results discussed in the previous section. O'Donnell and Winefordner (131) have reviewed the potential application of phosphorescence spectrometry in clinical chemistry, and a number of the drugs studied here have been previously studied by conventional phosphorimetry (132-135). Limits of detection for laser excited time resolved phosphorimetry are given in Table XV. With the exception of morphine and codeine, the detection limits using pulsed laser excitation and gated detection are 2 to 45-fold better than those previously reported. All limits of detection were calculated on the basis of S/N ratio of 3; the noise on the blank was taken as the standard deviation

PAGE 163

-156of 16 blank measurements. It is not likely that the noise was calculated in this manner in the literature data for morphine and codeine (132), and so previous detection limits should be treated with caution when compared with those determined in this study. With the exception of phencyclidine and anturane, the limits of detection in 0.75 M KI are not improved even though the phosphorescence signal has increased. The phosphorescence background with 0.75 M KI is larger than that of 10/90 v/v ethanol/water without 0.75 M KI, and the increase in signal does not offset the increase in background. The background luminescence is a broad structureless band centered about 500 nm. Phencyclidine is the exception because it has the largest increase in phosphorescence signal, and the background at 385 nm increases the least in 0.75 M KI. Phenobarbitol has a smaller increase in phosphorescence signal than phencyclidine when perturbed by iodide, so the detection limit does not improve. Tandearil and vinblastine are the only drugs which show a notable improvement in detection limit in 0.1 M AgNO^. For the currently available dyes for use in the CMX-4, the lowest wavelength which can be obtained by frequencydoubl ing is 265 nm. Using coumarin 504 as the laser dye, the peak output is at 270 nm. Changing the dye to sodium fluorescein or rhodamine 575 and tuning to higher wavelength gives less power than at 270 nm. For these reasons, no attempt was made to tune the laser as it was unlikely to give any improvement. The possible exception to this is vinblastine, but only 2-fold more power is available at any wavelength up to 365 nm than is available at 270 nm.

PAGE 164

Figure 20. Phosphorescence Spectra of Vinblastine Vinblastine concentration of 10 yg/ml . ( ) Vinblastine in 10/90 v/v E/W f = 15 Hz, t , = 0.5 ms, t = 10.0 ms, 1 x 10-7 A Full Scale. ° 9 (• • •) Vinblastine in 10/90 v/v E/W, 0.1 M AgN03 Recorded on Aminco-Bowman spectrofluorimeter with an Aminco-Keirs phosphoroscope attachment.

PAGE 165

-158b CO LlI UJ O UJ o CO Ui o X CL CO o X a. UJ > < 400 450 500 550 VP^/ELEMGTM (nm)

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159Table XV. Limits of Detection for Several Drugs by Laser Excited Time Resolved Phosphorimetry Emission Limits of Detection (pg/ml) in Compound^ Wavelength 10/90 v/v Ethanol/Water with (nm) 0.75 M KI 0.1 M AgNO^ 1.3 (0.1)^ 3.6 1.2 0.76 (6.0)^ 2.2 0.63 0.22 (0.01)^ 0.93 0.17 0.0072 (0.1)^ o.on 0.0025 (0.01)^ 0.014 0.091 0.039 (0.32)^ -0.028 (1.3)'^ 0.064 0.0080 (0.02)^ 0.044 (0.03)^ -0.034 0.024 0.043 0.18 0.19 0.12 -0.046 0.027 0.074 0.0057 0.0016 0.0017 0.0024 All drugs excited at 270 nm with CMX-4, 15 Hz, except Thiopropazate, excited at 337 nm with Avco Laser, 15 Hz. Based on Signal-to-Noise of 3, Observed Time Constant -^^3 s. Values taken from reference (132). Values taken from reference (133). Values taken from reference (134). Values taken from reference (135). Morphine

PAGE 167

•160Comparison of Excitation Sources In order to make a comparison of the nitrogen laser and flashlamp pumped dye laser, it is necessary to first discuss some of the differences pertinent to use in time resolved phosphorimetry. The CMX-4 has a longer pulse width ('x-l ys) than does the nitrogen laser ('v7 ns); the CMX-4 should do better in exciting phosphors due to their long lifetimes. On the other hand, the nitrogen laser has much larger peak power, but the power is only at 337.1 nm. Combining these factors, using Eq. IV. 1, and assuming: (i) the repetition rate of the lasers and complete detection system are the same; (ii) the comparison is based on a sample phosphor with a concentration low enough that the phosphorescence signal is directly proportional to the peak irradiance of the laser, P, and the molar absorptivity, e(40); (iii) the phosphorescence lifetime is greater than 100 ps. The ratio of the phosphorescence signal predicted for the nitrogen laser (S ) to the signal predicted for the CMX-4 (S-) is given by Sn , V337) t^ .,,(337) Because e(A) is a function of wavelength, it is necessary to specify a wavelength. At 337 nm for both assuming Pj^ = 40 kW (100 kW attenuated 40% by a filter), P^ = 300 W, and both are focused to the same area, the ratio is 0.93. If the CMX-4 is tuned to a different wavelength, the ratio will depend on P^(a) and c^{x) . Assuming 270 nm for the CMX-4 (P^ = 200 W), the ratio is 1.4 c(337)/e(270) . The molar absorptivity

PAGE 168

161must increase by more than a factor of 1.4 going from 337 to 270 nm for the CMX-4 to give a larger signal. For many molecules (for example, most of the drugs previously discussed), the first excited singlet is at higher energy than the energy at 337 nm and use of a nitrogen laser is not possible. For any molecule with a first excited singlet below this energy, either laser could be used to excite phosphorescence. The S/N ratio is of more interest than the signal alone, and so it is necessary to discuss the noise sources. To measure the noise due to variation in laser output, a short length of uranyl glass rod was placed in a sample cell and excited with each laser. The intense, shortlived, yellow-green luminescence was measured at 500 nm with a delay time of 0.2 ms, a gate width of 0.1 ms, and an OTC of 2 s. The luminescence of the uranyl glass was much more intense when excited by the nitrogen laser than when using the CMX-4, and so slit width and slit height were adjusted to give approximately the same detector photocurrent ('^>5 mA). The results for sixteen consecutive measurements are shown for the CMX-4 in Figure 21 and for the nitrogen laser in Figure 22. The detection system noise is negligible at these signal levels. It is readily apparent that the nitrogen laser is more stable in terms of pulse-to-pulse reproducibility indicated by the noise on the sixteen individual measurements. The long-term stability (time for each measurement was 1 minute) is also better for the nitrogen laser. The last measurement shown in Figure 21 is the signal obtained by repeaking the frequency-doubling crystal. The fact that signals returned to approximately the signal of the first measurement would indicate that this is the source of the slowly decreasing signal observed. This was a continuing problem in all experiments using the CMX-4.

PAGE 169

u CO o

PAGE 170

163riET-musy. t '^ _-»aer *:<^ AiLiSNauvii aAiiviaa J _j ID ZJ Z) Z) ZJ

PAGE 171

Figure 22. Noise on Nitrogen Laser f=15 Hz, td = 0.2 ms, tg = 0.1 ms, 5x10-6 A Full Scale, Uranyl glass luminescence at 500 nm.

PAGE 172

-165TIME > 0:1 1 n [] n y R [] 1 [1 f [] T n '

PAGE 173

•166Sample cell positioning has been a long standing problem in phosphorimetry. Hollifield and Winefordner (136) used an nmr spinner to rotate the sample cell and randomize positioning errors. This device works wery well in conventional phosphorimetry, but can not be used with pulsed excitation sources. Quartz capillary tubing is never straight enough to spin without wobbling even if the spinner itself does not wobble. Because the pulsed source is not on all the time by its very nature, it is possible for the sample cell because of sample wobble to be completely out of the optical path every time the source is triggered. The sleeve of the spinner assembly was used with a slip-fit Teflon cylinder that could be reproducibly positioned. The relative standard deviation, rsd, of 0.009 in the present studies is equivalent to rsd values for rotating sample cells. The different noise sources of importance are given in Table XVI. It is immediately evident that background variation from the solvent blank is the noise source of major importance. It is also the most difficult to reduce. Background variation is dependent on how the sample is cooled to liquid nitrogen temperature, along with nitrogen bubbling and ice crystals in the dewar. The optimum method was found to be cooling the sample for 20 s just above the liquid nitrogen and slowly lowering the sample tube into the liquid nitrogen. Limits of detection for benzophenone, quinine, phenanthrene, and carbazole are given in Table XVII. Benzophenone, phenanthrene, and quinine were chosen for the comparison of the CMX-4 and the nitrogen laser sources. The benzophenone signal and lifetime are unaffected by the heavy atom effect as its lowest triplet is n,TT*. The signal is 12-fold larger with the CMX-4, but the detection limits are only x.S times

PAGE 174

-167Table XVI. Noise Sources in Pulsed Laser Time Resolved Phosphor imetry Noise Origin Relative Standard Deviation CMX-4 Avco Source Induced 0.064 0.005 Sample Cell Holder Positioning 0.009 0.009 Sample Compartment Top Positioning 0.023 0.023 Solvent Background 10/90 v/v Ethanol/Water 0.14 0.15 (t = 5.2 ms) 10/90 v/v Ethanol/Water, 0.75 M KI 0.20 0.13 (t = 6.6 ms) 10/90 v/v Ethanol/Water, 0.1 M 0.075 0.067 AgN03 (t = 2.9 ms ) ^Relative Standard Deviation determined from standard deviation and mean of sixteen measurements.

PAGE 175

-168I X +-> E o i. T3 C 03 Sto en o s_ +-> o > CD c c o u OJ O) O O c o to •r™ i. O) Q 4-) I X CJ I— o CM ^ I— o o o CM r-. CO . . rCTl C\J l£> CM • >^ • 00 r— O 0)0)0) o o o > <: c OJ > o I/) a> CJ3 to B O) CM CM cyi ^ to O CM o O rCTl «* CM — o o Lf) CO CTl LO >;)r— CXD LD «;} OO tn CO o o o o o , '—I CD o ^^ — ' CD LU i>i eC CO CD . •— • CD CO o n o en o to to CO ro to O) Q O) E T3 o CL o C_) o o o CM CM CM o o o • • CM CM CM o o o . CM CM CM O O CM CM O O O ^ CM O CD "^ CM O O • I— I— O CM cr> cri o o; c o c QJ £= Q. o o N «^ OJ CO OJ E c •ILT) C r— •ILO CD0) c (U S-c E 4-> C c rtJ o c o OJ IT) OJ rE o c N m o -Q ^ s^ o c O cn c fO +-> c o u OJ E OJ > sO) to o O) to I o c en N o CM OJ o N to CL O) o X O) N N in tn o CM OJ to fO CO CO s_ QJ CO fO —I "^ I C X OJ 5: en (_) o SX O -1OJ to rtJ CQ fO o o u s_ > x: =3: o c o +-> (J OJ +-> O) o QJ O M to J3 s03 C_3 Ol c •r™ E sOJ -M OJ a o +-> a OJ to OJ to to o u > o OJ

PAGE 176

-169better due to an increase in the solvent background when excited at the lower wavelength. The detection limits are best in 0.1 M AgNO^ due to the smaller standard deviation of the background. Quinine is more efficiently excited at 337 nm, and so detection limits with the nitrogen laser are better. Due to the long lifetime of quinine, it is possible to gate off of the short-lived background in the ethanol/water. In 0.75 M KI, the background is smaller when excited at 337.1 nm than when excited at 270 nm which also makes the detection limits better using the nitrogen laser. In 0.1 M AgNO^ quinine gives a smaller signal using either laser than the phosphorescence signal from quinine in 0.75 M KI. The background and the background noise are also smaller in 0.1 M AgNO^ by a large enough factor to improve the detection limits. Going from the CMX-4 to the nitrogen laser increases the signal by a factor of 15, but only increases the background noise by a factor of 2. Time resolution is useful in 0.1 M AgNO^ as the quinine lifetime is longer than the background lifetime, but less useful in 0.75 M KI because the background lifetime and the quinine lifetime are close to the same value. Phenanthrene, like benzophenone, should give larger signals with the CMX-4 as e(337) = 225 and e ( 270) = 13000. Rather than a 40-fold increase in signal, only a 2-fold increase in signal was observed with the CMX-4 for phenanthrene in 10/90 v/v ethanol/water or 0.75 M KI and a 7-fold increase in signal for 0.1 M AgNO^, all relative to excitation with the nitrogen laser. This discrepancy is due to the difficulty in maintaining the frequency-doubling crystal at the optimum angle due to thermal drift, dye decomposition, and large variation in output energy from pulse-to-pulse. The value of 200 W for the peak power of the CMX-4

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-170is only an estimate based on manufacturers literature. A detector calibrated at 270 nm was not available to measure the actual laser output power. The detection limits for carbazole (carbazole was not used for a comparison compound) excited at 337 nm with the nitrogen laser are also listed in Table XVII. Less than 2-fold improvement in detection limit is found using 0.75 M KI, while nearly 15-fold improvement in detection limit is found for 0.1 M AgNO^. The initial intent of this work was to compare laser excitation with point source xenon flashlamp excitation. No results have been presented here using flashlamp excitation because of the experimental difficulties associated with reproducibly triggering the flashlamp. Even at repetition rates as low as 1 Hz, it was not possible to trigger the flashlamp 100 consecutive times without missing several flashes. This severely limited the analytical utility of these particular lamps. The cost ($125), lifetime (10 shots), and radio frequency noise associated with these sources does not make them particularly attractive in comparison to the Avco nitrogen laser. Conclusions Based on this study, the following conclusions regarding the external heavy atom effect and laser excited time resolved phosphorimetry may be reached: (i) Iodide and silver ions affect phosphorescence in a manner consistent with the external heavy atom effect; (ii) iodide and silver ions will increase the sensitivity of phosphorescence spectrometry for many molecules;

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-171(iii) iodide and silver ions may improve limits of detection in phosphorescence spectrometry depending on the magnitude of the increase of the background phosphorescence; (iv) the decrease in phosphorescence lifetime using silver ion in pulsed laser excited phosphorimetry makes more efficient use of the high peak power available from pulsed lasers, which may greatly improve both sensitivity and limits of detection; (v) the pulsed nitrogen laser is an excellent excitation source for phosphors having molar absorptivities as small as 10 at 337 nm due to its high peak power and excellent pulse-to-pulse reproducibil ity; (vi) the Chromatix CMX-4 flashlamp pumped dye laser requires the use of a ratio system to compensate for pulse-to-pulse variation, dye decomposition, and frequency-doubling crystal drift in order for it to be of great analytical use in phosphorimetry; (vii) the major noise sources in this study were found to be associated with immersion cooling in liquid nitrogen, so it is possible that the noise from these sources could be reduced by using conduction cooling.

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APPENDIX COMPUTER PROGRAMS USED FOR LIFETIME CALCULATIONS Two computer programs, written in BASIC, were used to calculate phosphorescence lifetimes on a PDP 11/20 minicomputer. The program FOS was used to calculate lifetimes from the slope of the least-squares line of the natural logarithm of the phosphorescence decay versus time. The program SLFOS calculates lifetimes by the same procedure after subtracting the signal due to a long-lived component. 172-

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-173P-iiUlUull t-U.hl I ."33 1 II.. i I", LOoO' 'i i-\-'.:,U ' ir ]:Lv.ft. Af-!D ^MtM Wf '.IWZ !.,';ru;. SlJKCPr MFLAY. S t OF" SWi:!! t; , INflJT 0" 4 rkir.T 'IF Yiii; j:)'-;i i!,':yr ^ ,-iJf ;,. c.-.m hum ):i. ankj . PiM riNi^ XNfur i' '.i 111. .'I '/F Hi m/ii-, ,',,ir .'.is'i'i;.. 'i 1 1: .'iia. itfii'OuY.i ;. y tuirii.iNi >jNfiir ?" '. jiJ! II ; 1 y pkii-n •]( YiU) iiu<.'i: ..Li'-TK-.a. ,m)i'il; with SAfii' i.uANK. inidi Y"\iNfiJi r? 1.6 IJi ^' J3'.'. l '\U"J '!,;00' 113 l.^•lm "iiAir: khijMoii.'i At 20 ll;JWT 'n.-.m-l 1:. 'MfJiiJT 1; I21! FiiKf •\ (Kiur !;-(M.vE:NT"\ifir-i)r i,i 2B l•l^l^J^ "ir.iT Ni.iiiTiFK'MNPin x* "iO II V.I-VI 3, 1 iirfj ;•; >'j'> riMisi "cM nfjis ji. L n.').u:''\.i:mi'Iii kj A'l ii;rNr "i.uLi r-'i [(• r IK i:i-,r:DMiiS"M.\'Fur G(Ji 6>'j iKiiJi "la-Li'iY ii,Ni:: tn r:;i::C:uNiii:)(HLY.NK) 'N input ikd o:j i-K-iMT -I. or <;i;iirf_pt;(«i..AKK) "MMrnT ckd /O i'U]:i< •CnXH LWAliP ( K.'.ANK' ) " \lfJ! 117 Gl 71 II(ij i Tlll-fi ;io ;':.' ii::.ii,'i "AN.'iiYTi: rii.L ,'«/Afif:. ' arJ.ur y% .'4 FKifri "awpepi iMi:: i;-( iitcofins'M: npi.it s<:!> r<'. Ml. 11 'uf.LAY lIMtIN !:iir(:f:Nr.<:,(AflALYTE) -MNPUT IK") Vt, PRjNr "NIIIU":irii OF SWFT.:.P'o M/y 70 IF VM ) IITLN V'4 ',•.' fvi (Ij ty( I )\n:' r(2('.'(ir'2 v:5 ;;;, ';,}! \f,i.>ri i 9 4 ni = f! (Ml i'Nl //',/) -"., (;-; ff;^ :nj.^^^ y v\ \ (.i )/c:i < i / :i'.)',, Ill y( I )-^'v'( I ) i /<; lOo I I 1 '.'( 1 ) vc I ) Nl iC/ I II YV YV(y< ! ) 1 ;)il i: Vk 1 ):' I .(,(' v)(;|-(..i •ilUN ]J7 I "/ Ml ','(1 . F'liu V'.:i ) , 1 iO 1(1 Y.l Yl l'.'< I ; 11.5 II r ', ^>^ Y/tv' I ) :>. I •..'< i 1 ; I I n; 1 n : ( I .:>). ri) i'.f.i i I I V \ ^ ;. ; M I :>'.) II I :••./. i , M M ) .;.> I i ( .' ,'M .". Ml II IjK M , -.'I I ) I '. / :i / . •"> nil ii I -i; I .'..I I-. >' I 1 I •// .1 : . .' 1 iiMi I .'.

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LIST OF REFERENCES 1. N. Omenetto, P. Benetti , L.P. Hart, J.D. Winefordner, and C. Th. J. Alkemade, S pec t roc him. Acta , 28B , 289 (1973). 2. J.W. Daily, Appl . Opt. . 15, 955 (1976). 3. J.W. Daily, Appl. Opt. , 16, 568 (1977). 4. G.D. Boutilier, M.B. Blackburn, J.M. Mermet, S.J. Weeks, H. Haraguchi, J.D. Winefordner, and N. Omenetto, Appl. Opt. , 17, 229 (1978). 5. D.R. de Olivares, Ph.D. Thesis, Indiana University, Bloomington, Indiana, 1976. 6. M. Hercher, Appl. Opt. , 6, 947 (1967). 7. M. Hercher, W. Chu, and D.L. Stockman, I.E.E.E. J. Quant. Electr. , QE4, 954 (1968). 8. F. Gires and F. Combaud, J. Phys. , 26, 325 (1965). 9. F. Gires, I.E.E.E. J. Quant. Electr. . QE2, 624 (1966). 10. R.W. Keyes, IBM J. Res. Develop. , 7, 334 (1963). 11. M. Yamashita. H. Ikeda. and H. Kashiwaga, J. Chem. Phys. , 63, 1127 (1975). "" 12. D.K. Killinger, C.C. Wang, and M. Hanabusha, Phys. Rev. A13 , 2145 (1976). 13. R.H. Rantel and H.E. Puntoff, "Fundamentals of Quantum Electronics," John Wiley, New York, N.Y., 1969. 14. J.J. Aaron and J.D. Winefordner, Talanta , 22, 707 (1975). 15. N. Omenetto, L.M. Eraser, and J.D. Winefordner, chapter in "Applied Spectroscopy Reviews," editor, E.G. Brame, Vol. 7 , Marcel Dekker, New York, N.Y., 1973. 16. R.J. Lukasiewicz, J.J. Mousa, and J.D. Winefordner, Anal. Chem. , 44, 963 (1972). ~~~" ~ 17. R.J. Lukasiewicz, J.J. Mousa, and J.D. Winefordner, Anal. Ch em., 44, 1339 (1972). ~ -179-

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BIOGRAPHICAL SKETCH Glenn D. Boutilier was born in Middletown, Connecticut, on February 7, 1953. After attending public schools in Maine, Ohio, and Massachusetts, he graduated from Shrev;sbury High School in Shrewsbury, Massachusetts. He obtained a Bachelor of Science degree in chemistry from Colorado State University in June, 1974. Graduate studies at the University of Florida were begun in September, 1974. He is a member of Phi Beta Kappa, the American Chemical Society, the Optical Society of America, and the Society for Applied Spectroscopy, -186-

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! certify that 1 have read this study and that in my cpiriicn it conforms to acceptable standards of scholarly preser.tauicn and is fully adequate, in scope and quality, as a disse-y^tation for the degree of . Doctor of Phriosophy. Vh^i i ja=^ V^ u. D' Winefordner/ Chairman graduate Research Professor of Chemistry I certify that I have read this study and that in iny opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Professor of Chemistry 1 certif/ that I have read this study ar.d that in my opinion it conforms to acceptable standards of scholarly presentation ar.d is Fully adequate, in scope and quality, as a dissertation for the degree of Doctor or Ptiilosophy. &.M. Schmid Associate Professor of Chemistry

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I c.fjrtify that I h6V(? read this study ?.nd that in ir.y opinion it cor.fornis to cicceptable standards of scholarly [•resentation and is fully, adequate, in scope and quality, a'a dissertation for the degree of / j Doctor of Pnilosophy. / / ^li::j:f^2g^' 3^~; ^ K.P. Li Assistant Professor of Chemistry I certify that 1 have read this study ^x\6. that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adooucite, in scope and quality. Doctor of Philosophy. as a dissertation for the degree of ^^_ H.A. Move Associate Professor of Food Science This dissertation was subiTiitted to the Graduate Tc-ulty of the Depfirtmsiit of Cheniistry in the College of liberal Arts and Sciences and to the Graduate Counci I reuuiroiierts for and was accepted as partial fulfillment of the the degree or Doctor of Philosophy. Decernber, 197S Dean, Graduate School

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