OPTIMIZATION OF SIGNALTONOISE 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
signaltonoise 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 SIGNALTONOISE RATIOS IN ANALYTICAL SPECTROMETRY. . . 4
Noise and SignaltoNoise 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
SignaltoNoise 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 SIGNALTONOISE 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 siqnaltonoise 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 2level and
3level 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,
7,8benzoflavone, 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 signaltonoise (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 (14)
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 (611). 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 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, 7,8benzoflavone, 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
SIGNALTONOISE RATIOS IN ANALYTICAL SPECTROMETRY
Noise and SignaltoNoise 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 lowfrequency (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 autocorrelation function of
the noise signals and when paired readings are considered; this treatment
yields general expressions for the signaltonoise (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 (2024) 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
tonoise 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
signaltonoise 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 jth 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 lowfrequency 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 wellknown. 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 lowfrequency 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 flameburner 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 wellknown
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 signaltonoise 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 autocorrelation
function and the spectral noise power involved.
The autocorrelation 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 autocorrelation function is simply a
deltafunction 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 RCfilter.
To obtain an expression of the noise in the frequency domain, use
can be made of the WienerKhintchine theorem (22,27), which relates the
autocorrelation 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 deltafunction, 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
correlationintime which makes the autocorrelation function of the
meter fluctuations due to the (originally) white noise differ from zero
also for T / 0. It also changes the autocorrelation 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 signaltonoise ratio, to the autocorrelation
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
signalplusbackground 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 signaltonoise 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
Sx 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 RCfilter
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 , tc *?"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 righthand side of the latter equation is
2
equal to ob which is the timeindependent variance of dxb(t). From the
2
very definition of the autocorrelation 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 autocorrelation 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 WienerKhintchine 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) measuringreadout
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; phaseshifts 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 signaltonoise 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 signaltonoise 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
Sn 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
{2Ki 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
4J
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 22 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/fdependence 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 22 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 lockin 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 00
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 noisefree;
(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 EW >
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 timeindependent autocorrelation 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 Tu+Ts
f du exp[(2u T)/Tr ] f exp(y/Tc)G(y)dy (11.50)
0 c Tu
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 22r fc 2 duexp(2u/Tc) f exp(s/T),G(s)ds +
Cs Tc2[I exp(T/T) 0 b u
's Tu+T
f du exp[(2uT)/lr] f s dy exp(y/ ),G (y)J (11.52)
0 Tu
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
iU+T
f* s ds exp(s/Tc [ (s) iG(s + T)] (11.53)
u
From Eq. 11.14, the WienerKhinchine 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 WienerKinchcine 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 signaltonoise
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
signaltonoise 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 signaltonoise 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 "quasistationary." 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
SignaltoNroise 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 signaltonoise 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 (3133). 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 excitationcontinuous 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 (sampleblank 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 ACexpression, 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 detectorelectronics 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 pulsingdetector 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 pulsingdetector 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 pulsingdetector
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 spectrometryagain 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
spectrometryin 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 sampleblank cycles. As the number of sampleblank 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
signaltonoise 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 SignaltoNloise 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 = sampleblank 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
WM1 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 sampleblank 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 SourceGated Detector Cases
Pulsed SourceGated DetectoraNo 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 SourceGated DetectoraWith 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 selfabsorption 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 deexcitation 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 deexcitation 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 deexcitation 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
radiativelyexcited, 2, electronic state to the ith vibra
1
tional level of the lower, 1, electronic state), s
k2 = nonradiative first order deexcitation 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 1012 C2 (Nm2 )
= Planck's constant, 6.626 x 1034 J s
2 2
= pure electronic transition moment, C n ;
= vibrational overlap integral (FranckCondon factor)
between vibrational levels in two electronic states
58
involved in the absorption and luminescence processes
(Q is vibrational coordinate); the BornOppenheimer
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
halfwidth 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 electronicvibrational 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 Ie2j(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 2level atom fluorescence radiance
(14).
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 3level 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 A1 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. YAl ( 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 2level 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
31
13
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
13
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
13
(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
12
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
21
12
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
21 1
12
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
13
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 Etype 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
21 '
12
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
3F 1 1 3
13
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)
31 1 1 32
13 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
21 1 )A +k +k l u
13 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 21 4ni "A +k +k 1 32
13 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) = (
P21 4
12
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 2level molecule, it is given by
Es(Oj)
10,2j
E* ni .,
12
2j33lO 'L AvV2
2 2j,10' "L 'AV 2j,10
I 1
(11 .67)
For a 3level 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 3level 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) ()
21
12
(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 106 W cm2Hz1
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 3level 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, a1 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 ,X24
76
3(2 2 2(a2a 3)
n3(t)= n3s exp(2 t) + (a23) 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 (4446) 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 (.0104) 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
3level 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 2level model. The
2level model predicts a saturation irradiance approximately 105 times
higher than the 3level 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 2level saturation
irradiance is greater than 107 times the 3level 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/ X24Y
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
4J
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 *
> CC\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 i11 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 in 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
4u 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 (24), 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 2level case under saturation conditions, the total
concentration, nT, can be determined by absolute measurement
of the steady state BFvalue, 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 Vterm occurring implicitly in the ifactor 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 3level molecule at room
temperature is 105 to 107 less than for a 2level atom or
molecule at any temperature or for a 3level atom or molecule
at high, e.g., flame, temperatures; because of the greater
halfwidths 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 2level
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 103 to 106 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
sourcegated 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 10fold better detection limits. The pulsed xenon
lamp had been predicted to give better detection limits (15). The con
tinuous wave source had an 85fold 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 (5456,60)
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 20fold 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 wideband 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 (6163). 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 (6870), photoacoustic spectrometry (71),
Raman spectrometry (72), and Coherent antiStokes Raman spectrometry
(73). Fixed frequency lasers such as the nitrogen laser (74), the
HeCd 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 (7779), 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 (8789) on
filter paper.
Rahn and Landry (90) found a 20fold 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 50fold. Other metal ions (Cd(II), Hg(ll), Zn(II), and
Cu(II)) have been studied as heavy atom perturbers (9192) at 77 K and
Ag(I) and T1(I) at room temperature on filter paper (19,9394).
Theory
The external heavy atom effect was first observed in 1952 by
Kasha (95) when the mixing of lchloronapthalene and ethyl iodide, both
colorless liquids, gave a yellow solution. The color was attributed to
an increase in the singlettriplet transition probability from increased
spinorbit coupling due to an external heavy atom effect. The increase
in spinorbit coupling was later proved by McGlynn et al. (96).
A spinorbit 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 spinorbit interaction
energy and inversely proportional to the energy difference between the
states being mixed (99). The spinorbit interaction energy for a
4
hydrogenlike 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 chargetransfer transition in a
chargetransfer 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 per
turber, P (102);
(iii) the tripletsinglet 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 chargetransfer
mechanism (i) or exchange mechanism (ii) is the most important. Some
investigators (100,104106) favor a chargetransfer mechanism while
others support the exchange mechanism (89,107112). There is excellent
