J. Phys. Chem. 1981, 85,949-953
pressure of 5.5 and 10.6 torr, respectively, and the longer the wavelength is the more important the isomerization processes are. The Absorption Spectrum. It is time now to come back to the absorption spectrum of the starting material and to try to draw some conclusions on the relationship between the behavior of the photoexcited state (photolysis) and the electronic state formed upon absorption of light (spectroscopy). It is, of course, tempting to link the isomerization of the monomer to the formation of Rydberg excited states. This simple view does not take into account all the likely internal conversion of electronic energy, from one excited electronic state to another or to the fundamental one. The results obtained here are far from sufficient to draw any valid conclusion. It is worthwhile to recall here a full discussion on the
949
different properties of P,T*, ~,R(3s), and P,U* excited states in olefins given recently by Kropp.26 The principal chemical property associated with the P,R* state of alkenes is cis-trans isomerization; the ~ , R ( 3 sstate ) undergoes a 1,2-methyl shift (skeletal isomerization); the positional migration of the double bond involves a [1,3]-sigmatropic hydrogen shift, and appears to be associated with neither P,T* nor a,R(3s) excited states.26
Acknowledgment. We express our gratitude to Professor C. Sandorfy (Universite de MontrBal) for his encouragements throughout this work. The help of the Department of Chemistry of the Universite de Montreal was also appreciated for the generous mass spectrometric analysis. (26) P. J. Kropp, US ARO Report, ARO-12810-2, 1978.
Resolution of the Fluorescence Lifetimes in a Heterogeneous System by Phase and Modulatlon Measurements Gregorio Weber Deparfment of Biochemistry, School of Chemical Sciences, University of Illinois, Urbana, Illinols 6 180 1 (Received: August 12, 1980)
A closed-form procedure is described for the determination of the decay constants and the relative contributing intensities of the N independent components of a heterogeneous fluorescence emission employing measurements of the phase shift and relative modulation of the total fluorescence at N appropriate harmonic excitation frequencies. At each frequency the phase and modulation measurements yield the real part of the Fourier transform of the fluorescence impulse response, G, and its imaginary part, S. It is shown that the moments of a distribution of the lifetimes are linear combinations of the Gs (zero and even moments) or the Ss (odd moments), and the rule for the construction of the coefficients of G and S in these linear combinations is derived. The classical de Prony method is used to obtain the lifetimes and fractional contributions of the components from the moments. For binary and ternary mixtures the numerical computations required are trivial. In the present state of the art, the lifetimes of the components of a binary mixture should be derivable with a loss in precision somewhat smaller than 1 order of magnitude with respect to the overall measured lifetimes.
Introduction The determination of fluorescence lifetimes by phase delay techniques goes back to Gaviolal (1927), and the effects expected in the overall phase and modulation by the presence of multiple fluorescence emissions were made clear by Dushinsky2 in 1933. In spite of these early beginnings no general method has been proposed to determine the proportions and lifetimes of the fluorescence components utilizing the phase delay and relative modulation data obtained a t different frequencies of the exciting light. This shortcoming has resulted in neglect of the phase techniques in favor of pulse fluorometry3 which often
permits resolution into components, although it is known that the phase methods are much faster in execution, and superior in precision in the measurements of overall decay, especially in the range of 100 ps to 3 ns.4 We present here an exact solution of the problem of determination of the proportion and lifetimes of N independent, noninteracting fluorophores, starting from the values of the phase shifts and relative modulation of the overall fluorescence excited a t N light-modulation frequencies. The numerical computations required are sufficiently simple to be performed in line with data acquisition. Although present-day precision may not be sufficient to
(1) E. Gaviola, 2. Phys., 42,85 (1927). Improved instrumentation of the same kind was used by W. Szymanowsky, 2. Phys., 95,460 (1936). Electronic detection of phase differences between photocurrents was introduced by E. A. Bailey and G. K. Rollefson, J. Chem. Phys., 21,1315 (1953) and A. Schmillen, 2. Phys., 135,294 (1953). Use of both phase and modulation to measure fluorescence lifetimes was first employed by J. B. Birks and W. A. Little, Proc. Phys. SOC., London, Sect. A , 66, 921 (1953). Cross-correlation techniques for phase and modulation measurements were introduced by R. D. Spencer and G. Weber, Ann. N.Y. Acad. Sci., 158,361 (1969). Continuously variable frequency of excitation has been realized by H. P. Haar and M. Hauser, Rev. Sci. Instrum., 49, 632 (1978). (2) F. Dushinsky, 2. Phys., 81, 7 (1933).
(3) Recent publications on pulse fluorometry that give an account of present state technology include the following: D. V. OConnor, W. R. Ware, and J. C. Andre, J.Phys. Chem., 83,1333 (1979);B. Valeur, Chem. Phys., 30,85 (1978); R. L. Lyke and W. R. Ware, Rev. Sci. Instrum., 48, 320 (1977); A. Gafni, R. L. Moolin, and L. Brand, Biophys. J.,15, 273 (1975). (4) R. D. Spencer and G. Weber, J. Chem. Phys., 52,1654 (1970); H. P. Haar and M. Hauser, under ref 1. (5) R. de Prony, J.Ec. Polytech. (Paris),1,24 (1795) appears to have used for the first time a method based on eq 27 to obtain the coefficients of dilation of an anisotropic solid. A modern description of Prony’s method is given by F. B. Hildebrand, “Introduction to Numerical Analysis”, McGraw-Hill, 1974, p 458.
0022-3654/81/2085-0949$01.25/00 1981 American Chemical Society
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The Journal of Physical Chemistry, Vol. 85, No. 8, 1981
Weber
tackle efficiently the resolution into more than two components, it is hoped that the exact character of the solution and the simplicity and speed of the numerical procedures will stimulate the development of phase fluorometry to make it a valuable technique of analysis of the fluorescence decays that make up a heterogeneous emission. It may not be out of place to reiterate here that the fluorescence from biological systems is virtually always heterogeneous and that resolution of the emission is often indispensable for a correct physical interpretation of any observed effects.
Resolution Theory Phase Delay and Modulation of the Composite Fluorescence. The addition of an arbitrary number of sinusoidally modulated components of the same frequency and variable amplitude and phase results in a sinusoidal wave of equal frequency, whose phase angle 9 and square amplitude hP are related to the component amplitudes ti and phases $i by eq 1 and 2. When the wave is a photan 9 = Cti sin $i/Cci cos 4i (1) i
L
M2 = (CcLsin $ J 2 i
+ (Ccicos $ i ) 2
(2)
1
tocurrent generated by the emission of fluorescence, 9 and M are respectively the angular delay and the relative modulation of the emission with respect to the values that characterize the harmonic excitation. The components are fluorescence emissions belonging to fluorophores with independent exponential decays, and in this case2v4
(3) cos $i where f i is the fraction of the total intensity of the fluorescence registered by the detecting element, usually a photomultiplier. By introduction of eq 3 into eq 1 and 2, we have 'i = f i
hP = S2 + G2
tan 9 = S/G
(4)
where
ai E WoTi (11) also a dimensionless quantity. With this notation N
G, = C f i ( 1 + p,2Q)-' i
S, = Cfip,ai(l i
i
S=
C f icos 4isin $i i
N
mk = CXiaik = YikG,
'
r,P = tan a r / w r
modulation of the total fluorescence relative to the modulation of the excitation, yields similarly an apparent lifetime by modulation given by eq 8. Equations 7 and rrM= ( M y 2- l)'l2/wr (8) 8 combined with eq 4 permit determination of the values G, and S, corresponding to a given frequency of excitation w,, from the experimental quantities M , and ar,namely
+, = [(l+ ( ~ , r , p ) ~ )+( l( ~ , r , ~ ) ~ ) ] -(9)~ / ~ S , = M , sin 9, = G,u,rT
(10) We consider here the general case in which the fluorescence is due to N independent components with lifetimes rl, ..., rN contributing fractions f l , ..., f N to the total detected intensity. If the fluorescence is excited with light modulated at N appropriate frequencies, we show below that an exact algebraic solution exists that permits the determination of the r and f values of the individual compo-
Y3kG3
( k = 0, 2, 4)
mk = YlkS1 + YzkS2 -k Y3kS3 (k = 1, 3, 5 ) (14) Equations 13 and 14 involve 18 different coefficients Yrk, since 1 1 r I 3 O I k 1 5
Replacing G, and S, by their values given in eq 12, in the right-hand side of eq 13 we have 3
=
3
fini(C yrk(1 + a?(p? + p?) + i=l r=l a t p s 2 ~ 2 ) ) ( k = 0, 2, 4) (15) 3
ni = rn (1 + p,2ai2)-1 =l
(6)
(7)
+ Y2kG2
(13)
(5)
At a fixed circular modulation frequency, w,, a measurement of the phase difference between the exciting light and the total fluorescence emission yields an apparent lifetime by phase, r;, given by eq 7. A measurement of M, the
+ p,2ai2)-l
The N measurements of G and the N measurements of S available as a result of observations a t N excitation frequencies are sufficient to determine the moments, of order zero to 2 N - 1, of a distribution of ai values. More precisely, the zero and even moments arise from linear combinations of the N values of G, while the odd moments result from linear combinations of the N values of S. Construction of the Moments of the Distribution of Lifetimes f r o m Phase and Modulation of the Total Fluorescence. Construction of the moments from the values of S and G is best discussed in connection with a given value of N . I shall use N = 3. If mk designates the moment of order k
mk
G = Cfi cos2 4'
G, = M , cos
nents from measurements of the overall phase delay and modulation of the fluorescence observed at the N excitation frequencies. If a suitable base frequency wo is chosen, w,/wo p r where p , is a small dimensionless number, and
(16)
r # s # t # r r, s, t = 1, 2 , 3 and from eq 12 and 14 3
mk =
3
fini(c Yrka$r(l + a?(p? + p i 2 ) + i=l r=l
(k = 1, 3, 5) (17) dp?p?)) By comparison of eq 15 and 17 with eq 13 and 14, we have X i = fini (18) =
3 r=l
Yrk(1 + a:(P:
+ p?) + atPs2pt2) ( h = 0, 2 , 4) (19)
=
3 r=l
Yrkaipr(1 + a;(Ps2
+ p?) + a?p,2pt2) (h = 1, 3, 5 ) ( 2 0 )
It will be noticed that f i is replaced by the product fini dependent upon ai but the same for all k's. The new distribution of lifetimes of which the moments are being constructed differs therefore from the original distribution in the weights assigned to each lifetime. Nevertheless, it is quite clear that, if the ai values are determined, the
Resolution of Fluorescence Lifetimes
The Journal of Physical Chemlstry, Vol. 85, No. 8, 198 1
original f i values can be easily recovered. When one writes eq 19 in the form r
r
ai4CY,kps2p,2 r
(k = 0, 2 , 4 ) (21)
it becomes apparent that for the equation to be valid the coefficients of terms with powers other than k must vanish, while the coefficent of uk must be unity. This condition gives
Cyrk = 6 k o r
Cyrk(p: r
+ pt2) = 6 k 2
= 6k4
cyrkP?P?
(22)
m2N-1
... +
Extension of the procedure of construction of the linear relations of G and S to give the moments, required for any number N of components, is immediate. By inversion of matrices like those in eq 24 and 25 of order N , one obtains coefficients Y r k which together with the G and S values determine the moments of the distribution of u,’s (eq 13 and 14). It will be noticed that all of these coefficients involve sums of products of the light-modulation frequencies of up to the power 2 N - 1. These frequencies are measured with very high precision, typically to 1 part in lo6,and the appearance of the high powers should not lead to experimentally appreciable errors; the errors in mk arise from simple addition of the errors in G or S, whatever the value of k. Computation of the Component Lifetimes from the ~ this purpose we inMoments by Prony’s M e t h ~ d .For troduce the symmetric products of the ai variables: el,e2, ...) These are formed by addition of the a’s taken singly, in products of two, in products of three, etc. We thus have el = Cui e2= Cuiul (i z j ) i
(i z j z k z i)
i
=
(26)
ala2.a.aN
The functions el,...) can be obtained from the set of 2 N moments as we shall presently show. Assuming that such values are known, one can form the equation aN - 8 1aN-l + 02aN-2- + ( - l ) N e N = 0 (27)
o
with 2 N - 1 Iq IN , and the null value of the sums is readily shown by setting aiq-lel = ais + aiQ-lCaj (j # i) I
~h
(25)
(28)
Since the 8 though not the m values are independent of the values of w chosen, each equation above involves addition of N separate equations in ai which must vanish identically. These N equations in ai have the form aiq - a p e l + (-i)Nq-NeN = (29)
ajq-383= aiq-’ Cajak
The two sets of nine equations (eq 22 and 23) may be written in matrix form:
i
..e
mN - mN.&
with 6’, = Kronecker’s delta, 0 I j I 5. In similar fashion we have from eq 20
O3 = Caia]ak
+ + (-l)NmN-leN = 0 + ... + (-l)NmoeN=O
- m2N-2el
95 1
+ ajq-3Cajakal
etc. The symmetric products of the a values are obtained by solving the system of equations in eq 28. For N = 3 the matrix form of this system would be
I
m4
-m3
m3 -m2 mZ
-ml
mz
;:1 I
mll m Q
(31)
=
Determination of the Proportions of the Components. Once the aivalues are determined as sketched above, it is a trivial matter to obtain the proportions of the components from the first N - 1 moments of the distribution (eq 16 and 17), which constitute now a system of linear equations in the f s with known coefficients. It may be noticed that, although the f s can be determined by employing fewer values of S and G than those available, all possible combinations of S and/or G values yield identical results. In other words, the f values are not overdetermined, and the procedure outlined above employs all of the experimental data to determine a unique set of lifetimes and contributing intensities. Binary and Ternary Systems Explicit Solution for Two Components. In this case observations of phase and modulation at two convenient excitation frequencies are required, and the equations developed for the general case become very simple. Except for a common divisor, p: - p12,which may be disregarded, the momenta of the distribution of lifetimes take the form
mo = P12G1 - P2Gz ml = P A - P2S2 m2 = -G1 + G2 m3 = - S l / p l + S 2 / p 2
(32)
The symmetric functions of the lifetimes become
el = (m3m0- m 2 m l ) / A
Ej2 =
(m3m1- m 2 2 ) / A (33)
with A = m2m0- m12and a1,a2= e 1 / 2 f ( e I 2 / 4- e2)lI2
(34) The relative intensities of the components may be calculated in many ways, all with identical results. One such is
.e.
the N roots of which are evidently al, u2,..,,aN,all positive quantities. Determination of the Symmetric Products of the Lifetimes from the Moments. Consider the N linear equations in the 0 s that result from the addition of the homogeneous j products of the form miel with 0 I i I 2 N - 1and 1 I I N
Explicit Solution for Three Components. This is not much more complex than the solution for two components. Let a1
= p22 - p32
a2
A3
= p32 - p12 E C 3 P?ai i=l
a3 = p12 - p2’ (36)
952
The Journal of Physical Chemistry, Vol. 85,No. 8, 1981
T1 = (a2 - m ) / [ m ( a l+ a2) - (ala2 - 111
The moments are given by 3
A3mk = ( - l ) k / 2C p:-k~iGi i=l
( K = 0, 2, 4)
(37)
3
A3mk = (-1)(k-1)/2 p t - k ~ i S i i=l
(k = 1, 3, 5) (38)
Calculation of Second Lifetime (r 2 )and Its Fractional Contribution to the Intensity ( f 2 ) , W h e n One of the Lifetimes (rl)Is Known. In many cases the fluorescence lifetime of one of the two components of the system is known, or may be independently determined by isolation of an excitation or emission region belonging to this component alone. In such cases the lifetime of the second component and the relative intensities can be obtained with less information: by measurements of phase and modulation at a single frequency, by phase measurements a t two frequencies, or by modulation a t two frequencies. We derive the relevant equations below: Determination of r2and f 2 from Phase and Modulation Measurements at a Single Frequency. In this case the experimental data permit calculation of S and G (eq 9 and 10) and, employing eq 3, we have G - cos2 41 = f2(cos2 4 2 - COS' 41) (39) S - cos dl sin = f2(cos 42 sin d2 - cos q51 sin 42) Eliminating f 2 from the last equations
p = (G- cos2 4,)/(S - cos 4, sin 41)= cos2 42 - cos2 $1 (40) cos 42 sin 42 - cos 41 sin 41 By setting al = wrl and a2 = or2,the last equation yields a2 = (8 + al)/(& - 1) (41) and f 2 is then calculated by eq 35. Determination of r2 and f 2 from Phase Measurements at Two Frequencies, w1 and up. When eq 1is applied to two components at frequency w1
T1 E tan G1 =
fl
cos2 $1 tan dl fl
cos2 $1
+ f 2 cos2 42 tan d2 + f 2 cos2 $2
(42)
+
Setting L1 E (Tl- a l ) / ( l a12),we obtain after replacing by 1 - f z (43) L1 = f 2 W l - (Tl - a z ) / ( l + a?)]
fl
At a frequency w2 = pol, L2 (T2- alp)/(l + a12p2)and Lz
f d L - ( T , - a2p)/(1+ a h 2 ) ]
(44)
Eliminating f 2 by means of the last two equations, we obtain the cubic equation
a23+ Cla22+ C2a2+ C3 = 0
(45)
with
C1 = -(LzT1p2 - LiTd/Co c 2
C3
Weber
= (L2 - LlP)/CO
= -(L2T1 - L I T ~ ) / C O
co = Lzp2 - L,p As eq 43 and 44 are symmetric in a, and a2,we expect the roots of the cubic to be al and a2and a symmetric function of them. Numerical solution with known values of 7 1 , 7 2 , and f2 shows that the roots are -al and -a2, and m = flap + f2al. Expressing T1and T2as functions of al, a2and m, eq 42 yields
T2 = p(a2 - m ) / [ p 2 m ( a l+ a2) - (p2ala2- 01 and from these it is readily shown that C3 = ala2m C1 = al + a2 - m
(46)
(47) As one of the roots of the cubic is the input quantity al, eq 45 is reduced to the quadratic a22
+ q1a2 + q 2 = 0
(48)
with q1 = C1 - al and q2 = C3/~1.The roots of the quadratic are a2 and m from which f 2 is readily obtained. Determinution of 7%aqdpf2from Modulation Measurements ut Two Frequencies. Let the modulations be M1 and M2,related to the modulation lifetimes T~~ 72Mby the equations
MI2 = 1 +
( ~ 1 7 1 ~ ) M2' ~
= 1+ (
~ 2 7 2 ~ ) ~(49)
where w1 and w2 = pwl are the excitation frequencies employed. When one applies eq 2 to a two-component system, after some elementary substitutions employing al = tan 41 and a2 = tan $2 f12 fi2 M12 = +1 + a12 1 + aZ2+ 2flf2(,
1 + ala2 +
a12)(l
+ a22)
(50)
and when one sets again m = flu2 + f2al,the last reduces to
M12 = (1 + m2)/[(1
+ a12)(1+ aZ2)]
(514
At the second frequency
M ; = (1 + m 2 p 2 ) / [ ( 1+ p2a12)(1+ p2aZ2)](51b) m2 is eliminated after setting B , 1 MI2(1+ a12) B2 M22(1+ p2a?)
(52)
giving
+ 1 - p2)/(B2- B1) = [(B1(1 + a2) - W2- a21/(al - a2)
~ 2 '= f2
P - ~ ( B -~ B2 '
(53) (54)
Precision of Measured and Resolved Lifetimes. The closed character of the solutions (eq 32-38) permits an easy estimate of the accuracy that is required in phase and modulation measurements to satisfactorily resolve a given binary or ternary system. One can start with the values of rp and rM that generate exactly the lifetimes of the components, round up these values to decreasing accuracy, and compute the degradation of the resulting lifetime values. It is found that binary systems can be resolved with a loss in precision of the generated lifetimes a little better than 1 order of magnitude if both components are present in appreciable proportions ( l / l O - l O / 1).Thus with Arp = ArM = f30 ps, we expect the resolved lifetimes to have a precision of h0.2 ns. Resolution of ternary systems requires considerably higher precision. Table I gives the range of precision in the resolved components when the common ratio of the lifetimes ( r 3 / r 2= 72/71) varies from 1.5 to 2.5, the three components are present in comparable proportions (0.25-0.401, and all lifetimes are longer than 2 ns. One evidently requires AT' = ArM = h5 ps to recover reliable values of T and f . The resolution of ternary systems by phase and modulation measurements is thus within the limits of the art, but its importance should not be exaggerated. In most cases, it would be more in the nature of a virtuoso performance than a useful asset because it does not seem likely for three components with appreciably different lifetimes to overlap so completely in
J. Phys. Chem. 1981, 85,953-958
TABLE Ia
common ratio 1.5 2 2.5 a
ps .
precision range O f T , ..., ns 0.5-3
0.1-0.8 0.1-0.7
Precision of lifetimes by phase and modulation
=
rt
5
both absorption and emission spectra that one of them could not be isolated or excluded by means of optical filters, reducing the problem of resolution to that of a binary system. Complementary Character of the Data from Pulse and Phase Fluorometry. As phase and modulation are measured under what may be called stationary conditions, all of the emissions, regardless of the time from excitation, contribute to the measured values of and M. On the other hand, the emissions following an exciting pulse after several fluorescent lifetimes represent a contribution too small to be accurately measured, and the evaluation of the emission a t very short times is limited by the finite instrumental response. The resulting uncertainties at the beginning and the end of the emission preclude the satisfactory closed-form solution that is available with the phase-and-modulation technique. We recall that G and S are respectively the real and imaginary parts of the Fourier transform of the fluorescence impulse response I ( t ) (eq 55). These relations G ( w ) = J m I ( t ) cos ut d t
(55) S ( w ) = - J m I ( t ) sin w t dt
953
suggest ways in which the results of pulse and phase fluorometry can be used to complement and confirm each other. From pulse fluorometry one has available a deconvoluted and truncated impulse response I ( t ) extending from t > 0 to t < a,and by numerical intergration values for G and S can be obtained for any particular frequency. On the other hand, values of G and S that include virtually all emissions from t = 0 to t = a can be computed with minimal propagated error from @ and M. These values from phase fluorometry can be expected to be much more accurate than the approximates from numerical integration of the experimental impulse response, and thus comparison of the two sets of values must permit a direct estimate of the errors owing to deconvolution and truncation in the latter method without regard to the number of components and other complexities of the system. Additionally, the values of G and S obtained from pulse fluorometry employing two or more frequencies could be subjected to the analysis here described to resolve the components of the decay. While the values of S and G from pulse fluorometry are less accurate than those from phase fluorometry, w can be chosen a t will to permit a more thorough analysis6than the one carried out with the two or three fixed frequencies commonly available in phase machines. It should be of considerable interest to compare such a method of resolution of the component lifetimes with others commonly employed for this purpose. (6) A method employing the Fourier transforms of the experimental impulse response (eq 55) and its fitting to expected values of the lifetime has been described by U. Wild, A. Holzwarth, and H. P. Good, Reu. Sci. Instrum., 48, 1621 (1977); O'Connor et al. (ref 3) point out that it does not seem to offer obvious advantage over the fitting to the impulse response itself.
Resolution of the pH-Dependent Heterogeneous Fluorescence Decay of Tryptophan by Phase and Modulation Measurements David M. Jameson and Gregorio Weber* Department of Eiochemistry, School of Chemical Sciences, University of Illinois, Urbana, Illinois 6 180 1 (Received: August 1.2, 1980)
The tryptophan fluorescence emission arises from the zwitterion and the anion, present in amounts determined by the pH of the solution. These forms interconvert in times much longer than the fluorescence lifetime, and their absorption and emission spectra are similar enough to make this an ideal binary system to test the resolution procedure by means of phase and modulation measurements at two excitation frequencies. Measurements were made by employing the excitation frequencies of 6,18, and 30 mHz, in the pH range 8-10,in which the relative zwitterion contribution varies from 0.82 to 0.09. Best resolution was expected and achieved by combining the data at 6 and 30 mHz. Resolved lifetimes were within f0.4 ns of the true lifetimes (3.1,zwitterion; 8.7, anion), and fractional contributions were within 10-2070 of expectancy. Such dispersion is predicted for phase and modulation measured lifetimes with standard deviations of -f50 ps, which in turn correspond to h0.15' phase error and f0.3% modulation error. With some limitations similarly good resolution was reached by fixing the value of one lifetime and employing fewer experimental data: phase and modulation at one frequency, phases at two frequencies, or modulation at two frequencies. For tryptophan no phase delay resulting from difference in energy of the exciting and fluorescence quanta was demonstrable, but correction for such effect may be, in general, needed, and a procedure for this purpose is described.
Introduction The preceding paper gave the theoretical treatment of the analysis of heterogeneous emitting systems by phase and modulation data. Here we describe the experimental verification of the theory and explore the precision and sources of error inherent in present-day instrumentation.
For this study we desired a well-defined, chemically heterogeneous system, i.e., a system wherein the lifetimes and fractional weights of each component are known with precision and in which the relative proportions can be varied in a kmwn fashion. To this end we have analyzed in detail the variation of the components in the fluores-
0022-3654/81/2085-0953$01.25/00 1981 American Chemical Society
