DEVELOPMENT OF A FLUOROMETRIC METHOD FOR STUDYING THE DURATION OF THE EXCITED STATE OF MOLECULES

A. M. Bonch-Bruevich

1. INTRODUCTION

For the study of intramolecular processes by luminescence methods, solutions of brightly fluorescing dyes with a quantum yield close to unity are favorable. Investigations of their luminescence provide important information about phenomena associated with spontaneous transitions of molecules from the excited to the normal state. On the other hand, changes in the structure of luminescing molecules, their concentration, the temperature of the solution, the physical properties of the solvent, the addition to the solution of molecules of another kind, etc., make it possible to trace the influence of intra- and intermolecular factors on the processes by which molecules pass from the excited to the normal state. It is further known that the study of emission spectra alone is insufficient for understanding the details of intramolecular processes and the features of the interaction of molecules in solutions. In this connection, S. I. Vavilov, in addition to “the spectrum in the ordinary sense of the word,” widely considered in his works the “spectra of yield, polarization, and duration of luminescence.”¹

Owing to the difficulties of direct observations of processes occurring over times of the order of \(10^{-8}\)—\(10^{-9}\) sec, a number of works were initially carried out on the indirect determination of the mean lifetime of molecules in the excited state. One such method is based on the observation of fluorescence quenching by foreign quenchers. The theory of this phenomenon, first given by S. I. Vavilov² and developed in later works³–⁷, establishes a connection between the change in luminescence yield, the mean duration of the excited state \(\tau\), the volume of the sphere of action of the excited molecule, and the probability of quenching upon encounter with a quencher molecule. Another method for determining \(\tau\) is based on observation of the degree of polarization of luminescence when the latter is excited by polarized light⁸, ⁹. Finally,

for this purpose one may use the increase in the degree of polarization of luminescence when it is quenched by foreign quenchers. In this case, as in the second indirect method, it is necessary to know reliably the molecular volume of the luminescent substance and the molecular viscosity \(^{7,10,11}\), which, as S. I. Vavilov pointed out, can be identified with the macroscopic viscosity only approximately \(^{12}\).

In a number of cases there are considerable discrepancies in the values obtained by direct and indirect methods. Thus, for example, by an indirect method it was found for a solution of rhodamine in water that \(\tau = 0.94 \cdot 10^{-9}\) sec, in ethyl alcohol—\(1.6 \cdot 10^{-9}\) sec, and in glycerin—\(2.7 \cdot 10^{-9}\) sec \(^{11}\). Direct measurements gave, respectively, \(2.5 \cdot 10^{-9}\) sec \(^{13}\), \(3.3 \cdot 10^{-9}\) sec \(^{14}\), and \(5.2 \cdot 10^{-9}\) sec \(^{13}\). A probable cause of the discrepancies may be insufficiently reliable knowledge of the molecular-kinetic characteristics of the solution. At the same time, if the values of \(\tau\) are reliably known from direct measurements, then the connection of \(\tau\) with such characteristics of the solution makes it possible to determine the latter experimentally or to test the physical conceptions underlying the corresponding theories.

The duration of the excited state of molecules can be determined directly by the pulse method or by the method of harmonic signals. The pulse method is used in studying the duration of scintillations and is based on measuring the time interval between the moment when an exciting particle (or \(\gamma\)-quantum) enters the phosphor and the moment when the intensity of the scintillation decreases by a definite number of times. Investigation of the shape of the luminescence pulse made it possible to show directly that the decay of the emission has an exponential character \(^{15}\). The pulse method has been used chiefly to examine phosphors of interest for scintillation counters; moreover, the values of \(\tau\) obtained in different works in many cases differ substantially for the same substances \(^{15–25}\).

For investigating the duration of fluorescence under photoexcitation, the method of harmonic signals is used. This method has also been used when exciting molecules by electron impact \(^{26,27}\) and by X-rays \(^{13,28}\). It is known that, for spontaneous radiation, the change with time of the light intensity \(I\) is described by the linear equations

\[ \frac{dI}{dt} - \frac{I}{\tau} = kE(t), \tag{1} \]

where \(k\) is a certain coefficient, constant for the given experiment. It follows from this that, under a harmonic variation of the intensity of the exciting light \(E(t)=E_0(1+m\cos\omega t)\) in the region \(t \gg \tau\) after the beginning of irradiation, the intensity of the luminescence light also follows

to the harmonic law \(I=I_0[1+M\cos(\omega t-\varphi)]\), where

\[ M=\frac{m}{(1+\omega^2\tau^2)^{1/2}}, \tag{2a} \]

and

\[ \varphi=\operatorname{arctg}\omega\tau . \tag{2b} \]

It is obvious that, in order to determine the quantity \(\tau\), it is sufficient to measure the ratio \(M/m\) or to find the magnitude of the phase shift \(\varphi\) between the exciting light and the luminescence light. With a more complex modulation of the light, relations (2) retain their validity for each harmonic of the exciting light (so long as equation (1) remains valid).

Usually, in fluorometers the quantity \(\tau\) is determined by measuring the phase shift \(\varphi\), which has a number of advantages over measuring the ratio between the modulation depths \(m\) and \(M\). Depending on the method of measuring this phase shift, fluorometers may be divided into photometric ones \(^{29—34}\), stroboscopic ones \(^{35—39,\ 26}\), and fluorometers with direct measurement of the phase shift \(^{13,\ 14,\ 27,\ 40—43}\). The latter are conventionally called “phase” fluorometers.

The accuracy of measurements on most fluorometers, as stated by their authors, lies in the range from \(3\cdot10^{-10}\) sec to \(10^{-9}\) sec. However, discrepancies in the values of \(\tau\) obtained in different works go far beyond these limits. Thus, for example, the data cited in the literature for stilbene under photoexcitation range from \(3.1\cdot10^{-9}\) sec to \(6\cdot10^{-9}\) sec \(^{13,\ 14}\), for phenanthrene—from \(5.2\cdot10^{-9}\) sec to \(13.5\cdot10^{-9}\) sec \(^{13,\ 15}\), for fluorene—from \(8.8\cdot10^{-9}\) sec to \(15\cdot10^{-9}\) sec \(^{13,\ 15}\), etc.

Large discrepancies in the values of \(\tau\) obtained for one and the same type of excitation considerably reduce the value of fluorometric measurements. These discrepancies may be due both to insufficient purity of the substances under investigation and to observational errors. Measurements by means of photometric and stroboscopic fluorometers have been subjected to critical evaluation in a number of works \(^{30,\ 38,\ 40}\). Recently, phase fluorometers, first developed in the laboratory of S. I. Vavilov, have become most widespread. The errors of measurements on them have not been considered, and only in a few works has attention been drawn to the possibility of systematic errors \(^{14,\ 41,\ 42}\).

At the same time, without an analysis of errors and the identification of the main sources of measurement errors, it is impossible either to increase the accuracy and sensitivity of modern fluorometers or to proceed to the study of weak luminescence. These two tasks are closely connected with each other, since the transition to weaker signals is accompanied by a decrease in measurement accuracy, while the sensitivity of devices with respect to the measurement of \(\tau\) increases with increasing brightness of the luminescence under investigation.

2. MODERN FLUOROMETERS

A summary of the principal characteristics of fluorometers constructed since the beginning of 1950 is given in Table I. A general survey of earlier fluorometers was given in the work of L. A. Tumerman^40. It is noteworthy that most fluorometers built in recent times are phase fluorometers. The modulation frequency is limited to 10–20 Mc/s, although with its increase the phase shift (for a given value of \(\tau\)) increases, which facilitates measurements. The reason for this lies in the difficulty of constructing high-frequency optical modulators. In most fluorometric setups, diffraction by standing ultrasonic waves, produced in quartz or, more often, in a liquid, is used for light modulation. An exception is the fluorometer of Birks and Little^34, in which the intensity of the radiation source itself is modulated. This is of considerable interest, since phase measurements with light modulation by means of diffraction on standing waves are complicated by a number of parasitic effects.

Of greatest interest with respect to circuit design, construction, and methodological procedures are the fluorometers of M. D. Galanin^14, Bailey and Rollefson^41, Schmillen^42, and Birks and Little^34. Since

Fig. 1. Diagram of the fluorometer of Bailey and Rollefson: \(L\) — light source; \(M\) — diffraction light modulator; \(1\) — generator feeding the modulator; \(K_1\), \(K_2\) — cuvettes with a scatterer and with the substance under investigation; \(2\) and \(3\) — phase shifters (uncalibrated and measuring); \(4\) — receiver; \(5\) — output indicator; \(6\) — auxiliary generator.

M. D. Galanin has described his apparatus in detail^14, we shall dwell here only on the last three fluorometers.

The scheme of the fluorometer of Bailey and Rollefson^41 is shown in Fig. 1; Fig. 2 shows the construction of the photoreceiving part of the instrument together with cuvettes and light filters. After passing through the diffraction modulator, the light from source \(L\) passes through two cuvettes (\(K_1\) and \(K_2\))—one filled with a liquid that strongly scatters light (a solution of sulfuric-acid barium), and the other filled either with a scatterer as well, or with a solution whose luminescence duration is being measured. The signals from the photomultipliers, at

Table I

) In another version, an X-ray tube with a modulating grid was used as the radiation source.
*) In another version, an electron gun with electron-current modulation was used.

which receive the scattered light and the luminescence light, pass through phase-shifting electrical circuits, and enter a radio receiver having a balanced antenna input. If the signals from both multipliers are equal in magnitude and in phase, they mutually compensate one another and at the output the signal falls to zero. By placing two cuvettes with scatterer in the path of the modulated light, with the aid of an uncalibrated electrical phase shifter and adjustment of the slits \(S_1\) and \(S_2\), located in front of the cathodes of the photomultipliers (Fig. 2), the instrument at the receiver output is set to zero. Replacing the second cuvette by a cuvette with the substance under investigation and introducing, by means of the measuring phase shifter, a known phase shift (and again equalizing the signal levels by adjusting the slits), the minimum readings of the output instrument are again established. The phase shift thereby introduced makes it possible to calculate \(\tau\) on the basis of relation (2a).

Fig. 2. Design of the photoreceiving part of the Bailey and Rollefson fluorometer: \(S_1\)—input slit; \(S_2\) and \(S_3\)—adjustable slits in front of the cathodes of the photomultipliers; \(F_1\) and \(F_2\)—light filters.

As phase shifters, \(RC\)-circuits are used, protected on both sides by cathode followers. A precision air capacitor with continuously variable capacitance is included in the measuring phase shifter. The phase shift produced by it was determined by calculation.

The use of a calculated phase shifter for measuring \(\tau\) cannot be considered successful, since its operation at a high modulation frequency is complicated by the action of parasitic parameters. As the authors themselves note, the phase shift depends on the parameters of the cathode-follower tubes and on the resistances in their cathode circuits. Therefore it was necessary to find experimentally the deviations from the calculated graph and to construct the corresponding correction curves, which complicates work with the instrument and reduces the accuracy of the measurements.

A substantial drawback of the Bailey and Rollefson fluorometer is also the chosen method for measuring \(\varphi\), complicated by the need to equalize, in the two channels, not only the phases but also the amplitudes of the signals. This, first, complicates and slows the measurements; and, since in phase-measuring devices with high-frequency optical signals there is always a more or less considerable instability of the reading, an increase in the measurement time corresponds to a decrease in the accuracy of determining \(\tau\). Secondly, in general one can hardly regard as successful a system requiring equalization of the signals, since this means that the strong signal must be attenuated to the level of the weak one. The latter is accompanied by a decrease in accuracy

measurements due to the action of fluctuation noise. To reduce it, in the Bailey and Rollefson fluorometer an additional signal from an auxiliary generator, tuned to a frequency differing from the light-modulation frequency by approximately 1000 cps (Fig. 1), is fed to the receiver antenna. After the receiver detector a narrow-band filter with a double T-shaped bridge is connected, which makes it possible to narrow the pass band of the device to 10 cps. However, when passing through the narrow-band system, the fluctuation signals give an alternating voltage resembling a harmonic signal, which is difficult to distinguish from the useful signal. Therefore, at different levels of fluctuation voltage in the two channels, additional systematic errors are possible. Finally, the very adjustment of the amplitudes of the signals by means of slits in front of the photocathodes of the photomultipliers is clearly unsuccessful, since this can easily lead to parasitic phase shifts owing to a change in the transmission time of the PMT signals when the illumination conditions of its photocathode are changed (see below).

Fig. 3. Schematic of Schmillen’s fluorometer: L—light source; M—cuvette with standing ultrasonic waves; F—light filter; N—investigated substance or scatterer; 1—resonant amplifier; 2—mixer; 3—narrow-band amplifier; 4—converter; 5—generator; 6—frequency doubler; 7—electromechanical frequency converter; 8—amplifier; 9—frequency multiplier; 10—oscilloscope tube.

The foregoing gives grounds to suppose that the measurement error indicated by the authors can be realized only under exceptionally favorable conditions. The fluorometer as a whole, however, cannot be regarded as a perfect instrument.

The scheme of Schmillen’s fluorometer^42 is shown in Fig. 3. The luminescence light, excited by radiation from a mercury lamp, modu-

modulated diffraction modulator, falls on the photocathode of the photomultiplier tube. The signal produced by the photomultiplier tube, after amplification, is fed to a mixing stage. To the second input of the mixer there is applied a signal whose frequency differs by 50 cps from the light-modulation frequency. To obtain this signal, the voltage feeding the modulator is combined with the voltage from the power line in the original electromechanical modulator. This modulator makes it possible to obtain a voltage with only one side frequency.

After the mixing stage a voltage of frequency 50 cps is obtained, and a change in the phase of the modulation of the light falling on the photocathode of the photomultiplier tube is accompanied by an equal change in the phase of this voltage. The voltage obtained is amplified by a selective amplifier and converted into a sequence of short pulses (following at intervals of 0.02 sec). These pulses are applied to an oscillographic tube with a circular sweep having a frequency of 200 cps. The position of the radial pulse on this sweep depends on the relation between the phase of the sweep voltage and the phase of the voltage coming from the photomultiplier tube. Passing from the scatterer to the luminescent substance, one can, from the displacement of the signal on the screen, measure the phase shift $\varphi$ connected with the duration of luminescence by relation (26).

Thanks to the transition to a low frequency and the use of a narrow-band amplifier, Schmillen considerably increased the sensitivity of his fluorometer for weak signals. This enabled him to advance into the region of small concentrations of the solution of the substance under investigation.

The principal drawback of Schmillen’s fluorometer is that, in the measurements, the reference phase is the phase of the voltage feeding the modulator. Meanwhile, a noticeable inconstancy is usually observed in the phase shift of the modulated light relative to the phase of this voltage, as well as fluctuations of the phase in different parts of the light beam emerging from the modulator$^{44,41}$. Further, the inconstancy of the frequency of the line voltage, when narrow-band amplifiers are present in the circuit, increases parasitic phase shifts. Finally, since the amplitude of the signals is not monitored and the circuit contains substantially nonlinear elements, amplitude-phase errors may occur$^{45}$. Unfortunately, Schmillen in his work does not pay attention to these sources of error, nor does he indicate whether measures were taken to ensure constancy of the illumination conditions of the photomultiplier cathode when passing from the scatterer to the substance under investigation. It is therefore difficult to estimate what systematic errors there may be in his instrument. Nevertheless, Schmillen’s fluorometer is of considerable interest, and the data obtained with its aid on the duration of the excited state of molecules should be regarded as more reliable than those found with the fluorometer of Bailey and Rollefson.

As was already stated above, in the Birks and Little fluorometer^34 the intensity of the radiation source itself is modulated. The scheme of this fluorometer is shown in Fig. 4. The radiation source is a gas-discharge gap formed by two sharpened tungsten electrodes sealed into a Pyrex bulb with a quartz window. A hydrogen pressure of 5 to 15 cm Hg is maintained in the bulb. The gas-discharge gap is included in an oscillatory circuit tuned to a frequency of 7.5 MHz and supplied by a 40 W generator. The voltage from this same generator, after frequency doubling, passage through a phase shifter, and amplitude limiting, is applied between the photocathode and the first dynode of the photomultiplier. The dc component of the photomultiplier current depends on the phase shift of the modulation of the light incident on the photocathode relative to the phase of the supply voltage, and reaches a maximum when they are in phase. During measurements, the dependence of the dc component of the multiplier current on the phase shift introduced by the phase shifter is recorded, first when the photocathode is irradiated by light from the radiation source, and then when the substance under investigation and crossed light filters are placed between the radiation source and the photocathode. The phase difference corresponding to the maxima of the two curves makes it possible to find \(\tau\), using relation (2a). The phase shifter is an artificial electrical line with a movable coil, graduated by lengths of cable with known signal delay time.

Fig. 4. Scheme of the Birks and Little fluorometer: \(L\)—gas-discharge radiation source; \(F_1\) and \(F_2\)—light filters; \(N\)—sample of the substance under investigation; 1—generator; 2—frequency doubler; 3—delay line (reference phase shifter); 4—amplifier-limiter; 5—dc instrument.

In its principle of operation the Birks and Little fluorometer repeats the Hottel apparatus^46 and a number of other devices of analogous type. Its accuracy is relatively low, measurements on it are complicated, and their results may be substantially distorted by amplitude-phase dependences and by harmonics of the fundamental frequency in the spectrum of the intensity modulation of the exciting light. As already noted, the main interest in the Birks and Little fluorometer is the modulation of the intensity of the radiation source itself. However, the absence of any sufficiently exhaustive data on the properties of such a source, its brightness, the stability of the phase relations between the emitted light and the supply voltage, etc., makes it difficult to evaluate this method from the standpoint of fluorometric measurements.

3. MAIN SOURCES OF MEASUREMENT ERRORS IN PHASE FLUOROMETERS

Regardless of the design of the phase fluorometer, the measurement on it of the fluorescence lifetime is reduced to a twofold determination of the phase difference between the signals in two channels—when a scatterer is placed in the measuring channel and when the scatterer is replaced by the substance under study. In most fluorometers the measurements are made by the null method, for which the instrument includes a device permitting the introduction into one of the channels of a standard phase shift, while the phasemetric device is used as a null indicator. Such a standard phase shifter in the fluorometer of M. D. Galanin is a variable optical path, the length of which can be varied by 6 m; in the fluorometer of Bailey and Rollefson it is a graduated \(RC\)-circuit; in the fluorometer of Birks and Little it is an artificial electrical line, etc.

The accuracy of measurement of the mean duration of the excited state of molecules is substantially determined both by the correct operation of the standard phase shifter and by the resolving power \(\Delta\varphi\) of the phasemetric device. The latter, in turn, is determined by the resolving power \(\Delta\varphi_0\) of the phase indicator and can be increased both by going over to a more sensitive indicator and by introducing special devices into the pre-indicator part of the phasemeter, having the meaning of “amplifiers” of phase shifts\(^ {45}\). However, as will be clarified below, increasing the resolving power of the indicator beyond a certain limit may be irrational, both because of the action of fluctuation noise and because of specific errors of phase measurements with optical signals.

Fig. 5. Generalized graph of the relative error of measurements on a phase fluorometer.

Noting that the value \(\tau\) is computed from the phase shift found as a result of a twofold phase reading, from expression (26) we determine the relative error of the measurements

\[ \frac{\Delta \tau}{\tau} = \frac{1+\omega^2\tau^2}{\omega\tau}\,2\Delta\varphi_0 . \tag{3} \]

In Fig. 5 a generalized graph is given of the relative error of measurements

\[ \frac{1}{2\Delta\varphi_0}\cdot \frac{\Delta\tau}{\tau}=f(\omega\tau). \]

It is easy to see that the quantity \(\Delta\varphi_0\) determines the minimum error-

accuracy of measurements:

\[ \left.\frac{\Delta\tau}{\tau}\right|_{\min}=4\Delta\varphi_{0} \quad \text{for } \omega\tau=1, \]

and, together with the modulation frequency, determines the limits of the values of \(\tau\) within which the quantity \(\Delta\tau/\tau\) does not exceed the specified value \((p_{0})\). This region is bounded by the quantities

\[ \omega\tau^{*}_{1,2} = \frac{p_{0}}{4\Delta\varphi_{0}} \left[ 1 \pm \sqrt{1-16\left(\frac{\Delta\varphi_{0}}{p_{0}}\right)^{2}} \right]. \tag{4} \]

In Fig. 5 the values \(\omega\tau^{*}_{1}\) and \(\omega\tau^{*}_{2}\), corresponding to \(\Delta\varphi_{0}=1^\circ\) and \(p_{0}=0.2\), are marked. In Fig. 6 a family of curves \(\omega\tau^{*}_{1,2}=f(\Delta\varphi_{0},p_{0})\) is given, where the quantity \(p_{0}\) is a parameter. On the ordinate axis,

Fig. 6. Family of curves \(\omega\tau^{*}=f(\Delta\varphi_{0},p_{0})\), determining the region of values of \(\omega\tau\) and \(\tau\) within which the measurement error does not exceed the specified value \(p_{0}\).

in addition to the values of \(\omega\tau^{*}\), the quantities \(\tau\) (in microseconds) are also plotted for the modulation frequency \(F=10\) MHz. In many fluorometers an oscillographic method is used as the phase indicator.

tube. Its resolving power is about \(1^\circ\). In Fig. 6 a dashed line is drawn corresponding to this resolving power. The intersections of this straight line with the graphs corresponding to definite values of the quantity \(p_0\) make it possible to judge visually the working region of the instrument.

The resolving power of the phase indicator is substantially determined by the level of fluctuation noise, the main sources of which lie in the photomultipliers and in the electron-tube circuit of the pre-indicator part of the phasemeter. In phase measurements, fluctuation noise leads to fluctuations of the magnitude of the measured phase difference between two signals. As a result, measurements, for example with the aid of an oscillographic tube, are complicated by blurring of the pattern observed on its screen and, accordingly, by a decrease in the accuracy of the reading. Therefore fluctuation oscillations of the phase are, in a certain sense, equivalent to a deterioration of the resolving power of the indicator. We have specially considered the question of the action of “phase noise” on the resolving power of phasometric devices\(^{45}\). The fluctuation voltage may be represented in the form

\[ U_{\mathrm{ш}} = E \cos \Phi \tag{5} \]

(\(E\) and \(\Phi\) are random independent quantities), i.e. as a harmonic signal randomly modulated in amplitude and phase\(^{47}\). To determine the threshold sensitivity of the device in phase measurements (by analogy with threshold sensitivity in amplitude measurements) one may use vector diagrams. For this purpose it is convenient to represent expression (5) in the form \(U_{\mathrm{ш}} = E \cos(\omega t - \theta)\) and to estimate the phase noise by the angle \(\varphi_{\mathrm{ш}}\) between the signal of frequency \(\omega\) and the resultant of the vectors \(E\) and the vector of the signal \(U_c\) (Fig. 7). Proceeding in this way\(^{45}\), one can show that, in every case, if the level of fluctuation noise remains below the level of the useful signal, the threshold sensitivity of the device

Fig. 7. Vector diagram for considering fluctuation noise in phase measurements.

\[ \Delta \varphi \simeq \Delta \varphi_0 + \Delta \varphi_{\mathrm{ш}} \simeq \Delta \varphi_0 + a \frac{u_{\mathrm{ш}0}}{U_c}\sqrt{\Delta f_{\mathrm{п}}}, \tag{6} \]

Here \(a\) is a certain coefficient, \(u_{\mathrm{ш}0}\) is the effective value of the fluctuation voltage referred to a unit spectral interval, and \(\Delta f_{\mathrm{п}}\) is the passband of the measuring device (defined as the band, equivalent with respect to the change in the energy of a signal with a continuous spectrum, of an ideal filter with a Π-shaped transmission characteristic). For \(\Delta f_{\mathrm{п}} = 5\) kc, there were experimentally obtained

the following values of the resolving power of the oscillographic tube as a phase indicator:

Using the graphs in Fig. 6, one can judge how strongly the error in determining $\tau$ increases and how the working range of the fluorometer narrows even at a comparatively small level of fluctuation noise. With a decrease in the fluorescence intensity, the accuracy of measuring its duration on a fluorometer with an oscillographic tube drops sharply. As follows from expression (6), by narrowing the pass band of the phase-measuring device one can, in principle, reduce the value of $\Delta\varphi_{\mathrm{ш}}$ arbitrarily and thereby increase the resolving power of the fluorometer up to the resolving power of the phase indicator. However, in doing this it is necessary to take into account that, as is known, the phase shift of a signal on passing through any selective electrical circuit depends on the signal frequency. Therefore the total error in measuring the phase shift is

\[ \Delta\varphi=\Delta\varphi_0+\Delta\varphi_{\mathrm{ш}}+\Delta\varphi_{\omega} \approx \Delta\varphi_0+a\,\frac{u_{\mathrm{ш}0}}{U_c}\sqrt{\Delta f_{\mathrm{п}}}+\chi\Delta\Omega, \tag{7} \]

where $\chi=\left|\dfrac{\partial\varphi}{\partial\Omega}\right|_{\Omega=1}$ is a coefficient determined by the form of the electrical circuit and the width of its pass band,^45 and $\Omega$ is the relative instability of the frequency of the amplified signals, determined in the fluorometer by the instability of the light-modulation frequency and of the frequency of the auxiliary generator in the phase-measuring part.

The higher the selectivity of the electrical circuit, the greater, generally speaking, the value of $\chi$, and the reduction of $\Delta\varphi_{\mathrm{ш}}$ is accompanied by an increase in $\Delta\varphi_{\omega}$. Therefore, strong narrowing of the pass band of the pre-indicator device in ordinary installations is irrational. Without dwelling here on details, we note that frequency-phase errors can be appreciably reduced by constructing a symmetric two-channel phase-measuring device, which can naturally be done in a fluorometer with two optical channels. Another way is stabilization of the frequency of the signals entering the amplification paths of the fluorometer. In this respect, a double frequency-conversion system is of great interest, making it possible, independently of the signal frequency, to fix with high accuracy the intermediate frequency at which the main amplification is carried out.^45 Finally, one may abandon strong narrowing of the pass band of the pre-indicator device, switching, in order to reduce the influence of fluctuation noise, to an inertial phase indica-

tor. Such an indicator may be an ordinary pointer instrument with an inertial \(RC\)-circuit, included in this or another phase-sensitive circuit. An additional advantage of an inertial phase indicator is that its resolving power may be higher than that of an oscillographic tube.

The results of fluorometric measurements may be substantially distorted by systematic errors. There are several sources of such errors in phase fluorometers. First, a change in the illumination conditions of the photocathode of the photomultiplier during measurements can lead to appreciable errors.

Fig. 8. a) Experimental graphs of the change in the flight time of electrons in the FEU-19 when the light spot is displaced along the diameter of the photocathode parallel to the generatrices of the dynodes (curve 1) and perpendicular to the generatrices of the dynodes (curve 2). b) Graphs of the change in the flight time of electrons as a function of the illumination of the photocathode (photocurrent magnitude) at different supply voltages of the FEU.

We carried out a special examination of the dependence of the electron flight time in the FEU on the place of illumination of its photocathode and on the magnitude of the photocurrent. Fig. 8 gives experimental curves for one of the multipliers. From their consideration it is evident that a change in the illumination conditions of the photocathode during the measurement process can lead to errors in determining \(\tau\) of several millimicroseconds, which may exceed by one or two orders the errors due to the resolving power of the indicator. Such a change in illumination conditions is quite possible both when replacing the scatterer with the specimen under investigation and when introducing an additional phase shift, by means of an optical line of variable length, into one of the channels of the setup. So far as we know, no attention has been paid to these errors.

The second source of systematic errors may be differences in the phases of modulation in different parts of the light beam emerging from the optical modulator, which we found in studying diffraction modulator 4. These phase shifts are such that they are equivalent to several millimicroseconds of the measured

my value \(\tau\). Moreover, as subsequent observations have shown, the phase of the modulation can depend noticeably on the wavelength of the modulating light. Therefore, if different parts of the beam are used in placing the scatterer and observing the fluorescence, in the geometrical or spectral sense, then a large systematic error in the determination of \(\tau\) is possible. As far as we know, attention has been paid to phase differences in different parts of the light beam in only one paper \(^{41}\); no attention at all has been paid to phase differences in different parts of the spectrum.

In addition to what has been said, we note that differences in the modulation phases in different parts of the light beam change with time. Therefore, with imperfect light division between the two channels, a considerable instability of the reading is observed, which further decreases the accuracy of the measurements. The nature of the fluctuations of the reading under poor and good light division is illustrated by the experimental plots given in Fig. 9.

Fig. 9. Experimental plots of fluctuations of the reading of the fluorometer indicator with poor (1) and good (2) light division in the optical part of the instrument.

Finally, substantial systematic errors may be associated with the operation of the electron-tube part of the fluorometer. Errors are caused by parasitic couplings between the channels and within the channels, as a result of which, when the phase or level of the signal changes in one part of the setup, the phase of the signal in another part is changed parasitically. More substantial errors may be due to the dependence of the transit time through the electron-tube

depends on the signal circuit from its amplitude, to which we drew attention when considering questions of phase measurements^45. In this case the measured quantity \(\varphi\) will depend on the intensity of the fluorescence. In addition to an incorrect determination of \(\tau\), this may lead to substantial distortions in the character of the dependence of the fluorescence duration on various factors that change the brightness of the glow.

Being nonlinear, amplitude-phase distortions are connected with the presence of nonlinear elements in the circuit. However, as is known, when a harmonic signal passes through a nonlinear system or through a sequence of several nonlinear systems and the spectrum of the signal is correspondingly complicated, the phase of the component of the fundamental frequency remains unchanged. This corresponds to the fact that a harmonic signal subjected to a nonlinear transformation continues to be described by a symmetric function of time. Therefore, for an amplitude-phase effect to arise, it is necessary that the signal cease to be symmetric; this may occur with a combination of linear and nonlinear distortions. A combination of the two kinds of distortions leading to amplitude-phase effects may take place both in a single element of the device and in a sequence of several elements.

A system in which linear and nonlinear distortions leading to amplitude-phase dependences are combined is an oscillatory circuit whose inductance is made with the use of iron or another magnetic material (core, shield). A change in its magnetic permeability as a function of the magnetic-field strength leads to a change in the natural frequency of the circuit when the amplitude of the signals changes (nonlinear effect). The latter, in turn, owing to the finite value of the quantity \(\varkappa\) (linear effect), is accompanied by a change in the phase of the signal at the output. Consequently, in this case the amplitude-phase distortions are substantially determined both by the amplitude instability of the magnetodielectric and by the value

\[ \Delta\varphi_U=\left.\frac{\partial\varphi}{\partial U}\right|_{U=U_0}\Delta U =\left.\varkappa\frac{\partial\Omega}{\partial U}\right|_{U=U_0}\Delta U. \tag{8} \]

As an illustration of how strongly the dependence of the phase shift of a signal passing through a resonant amplifier on the voltage across the circuit can be expressed, Fig. 10 gives an experimentally obtained graph \(\Delta\varphi_U=f(U)\), relating to the use in the circuit of a pot core made of carbonyl iron \((HF—B_1)\). Depending on the type of core, its construction, the coefficient of amplitude instability of the magnetodielectric, etc., the quantity \(\dfrac{\Delta\varphi_U}{\Delta U}\) may vary within considerable limits.

Another nonlinear element with which amplitude-phase distortions may be associated is an electron tube with a resonant (complex) load. If the tube operates in an overvoltage-

nominal mode, and the circuit is inaccurately tuned to the signal frequency, then the harmonic voltage acting on the tube grid is transformed into a voltage described by an asymmetric function of time.

In order to avoid amplitude-phase errors, special measures must be taken in constructing a fluorometer, measures that are usually not taken in the construction of other electron-tube measuring devices. In addition, the signal level must necessarily be carefully monitored, and the signals from the phosphor and from the scatterer must be equalized.

Fig. 10. Phase-amplitude dependence for an oscillatory circuit with an armored core made of carbonyl iron.

The above consideration of the errors of fluorometric measurements suggests that the accuracies of measurements on fluorometers reported in the literature are overestimated. Usually the resolving power of the instrument for strong signals is given as the accuracy of measurement. The possibility of a number of systematic errors is not considered at all. In this connection, the discrepancies in the values of $\tau$ reported in the literature for different substances may to a considerable extent be attributed to measurement errors.

4. A NEW PHASE FLUOROMETER

The analysis of errors of fluorometric measurements set forth above was carried out in connection with the development of a new phase fluorometer. The results of this analysis made it possible to take a number of measures to reduce measurement errors, increase their reliability, and increase the sensitivity of the fluorometer.

The general scheme of the new phase fluorometer is shown in Fig. 11. For modulating the light, a diffraction modulator with standing ultrasonic waves in a liquid (xylene), operating at a frequency of about 12 MHz, was used. A special design of the rigid mounting of the piezoquartz plate exciting the ultrasonic oscillations made it possible to obtain a stable pattern of ordered standing waves of compression and rarefaction in the liquid44. In contrast to the usual practice, the modulator is constructed so as to select not the central maximum, but all the lateral maxima of the diffraction pattern; this makes it possible to eliminate the significant constant component of the exciting light falling on the substance under investigation. Owing to this, the level of fluctuation noise produced by the photomultiplier tube is reduced.

In the instrument, measures have been taken for proper, in the geometrical and spectral sense, light separation between the measuring channel

and the comparison channel. A filter that selects one or another region of the spectrum of the exciting light is placed before the beam-splitting plate; the latter is neutral, and, finally, the geometry of the setup is such that the light beams after splitting are not limited.

The specimen of the substance under investigation and the diffuser replacing it are arranged in such a way that the illumination of the photocathode of the photomultiplier remains unchanged when passing from the diffuser to the substance under investigation. Likewise, changing the length of the optical line in the second channel does not change the illumination conditions of the photocathode of the second multiplier. For the fluorometer, multipliers are selected with a minimal dependence of the signal-transmission time on the point of illumination and on the intensity of illumination of the photocathode. In addition, only the central part of the photocathode is selected by a diaphragm. The construction of the fluorometer makes it possible to verify easily the absence of parasitic effects in its optical part.

Fig. 11. General layout of the new phase fluorometer: \(Л\)—source of exciting light (СВДШ-250); \(щ\), \(к\), and \(ш\)—elements of the optical modulator; \(\Phi_1\) and \(\Phi_2\)—light filters; \(Д\)—neutral beam-splitting plate; \(O\)—neutral attenuator; \(P\)—diffuser; \(N\)—specimen of the substance under investigation or diffuser; \(З_1\), \(З_2\), and \(З_3\)—a mirror system allowing the length of the optical path in the comparison channel to be varied smoothly by \(30\) cm; \(1\)—generator; \(2\)—phasometric device; \(3\) and \(4\)—phase-shift indicators (oscillographic tube and phase-bridge instrument).

The signals from the photomultipliers arrive at the two inputs of a symmetrical two-channel phasometric device (Fig. 12), used as a null indicator. The phasemeter has two phase indicators—an oscillographic tube and a special phase-sensitive bridge (phase detector) with a pointer instrument, the readings of which depend on the phase shift between the signals acting at both inputs of the phasemeter. The duration of luminescence is determined on the basis of the relation

\[ \tau=\frac{1}{2\pi F}\operatorname{tg}\varphi, \tag{9} \]

where \(F\) is the frequency of light modulation, and \(\varphi\) is the change in the phase of the signal delivered by the photomultiplier in the measuring channel when the substance under investigation is placed there instead of the scatterer.

When measuring the angle \(\varphi\), the main indicator used is a phase detector, while the oscillographic tube serves for adjustment and checking of the operation of the setup, as well as for rough measurements and a visual judgment of the level of fluctuation noise.

Fig. 12. Block diagram of the fluorometer phase-measuring device: 1 and 2—high-frequency amplifiers; 3—heterodyne; 4—tube voltmeter; 5 and 6—buffer stages; 7 and 8—mixers; 9 and 10—calibrated phase shifters; 11 and 12—blocks of controlled amplifiers of the APA system; 13, 14, 17, and 18—intermediate-frequency amplifiers; 15 and 16—uncalibrated phase shifters; 19, 20—cathode followers; 21, 22, 27, and 28—output stages; 23 and 24—detectors of the APA system; 25 and 26—paraphase stages; 29—phase bridge; 30—pointer instrument in the phase-bridge circuit; 31—oscillographic tube.

The resolving power of the phase detector is chosen equal to \(0.1^\circ\) (of the order of the zero-reading instability present in the instrument). The use of an inertial indicator (phase detector) makes it possible to retain high sensitivity in the presence of fluctuation noise. This is illustrated by the experimental data of Table II. \(\left(U_{\mathrm{sh}} = u_{\mathrm{sh}0}\sqrt{\Delta f_n}\right.\) is the effective voltage at the output of the pre-indicator device.)

Table II

In Fig. 13 a family of graphs \(\omega\tau = f(\Delta\varphi_0, p_0)\) is shown, on which vertical straight lines have been drawn corresponding to the resolving powers of the indicators for different values of the ratio \(\dfrac{U_{\mathrm{sh}}}{U_c}\). The intersections of these straight lines with the corresponding graphs

Fig. 13. Family of graphs \(\omega\tau = f(\Delta\varphi_0, p_0)\) and straight lines corresponding to the resolving powers of the phase indicators of the constructed fluorometer for different values of the ratio \(\dfrac{U_{\mathrm{sh}}}{U_c}\). Dashed vertical straight lines—1, 2, 3, 4—for the phase detector, respectively at \(\dfrac{U_{\mathrm{sh}}}{U_c}=0,\ 0.25,\ 1.0,\) and \(2.0\); solid straight lines—5, 6, 7, and 8—for the oscillographic tube, respectively at these same values of the ratio \(\dfrac{U_{\mathrm{sh}}}{U_c}\).

determine the region of values of \(\omega\tau\) and \(\tau\) within which the measurement error does not exceed the specified value \(p_0\).

In constructing the phasometric device of the fluorometer, special measures were taken to eliminate the possibility of systematic errors in measuring the value \(\tau\) associated with its operation. In addition to the appropriate design of the vacuum-tube part of the fluorometer, the choice of suitable tube operating conditions, careful shielding of individual units, etc., automatic regulation of the signal amplitude (ARA) was introduced into the phasemeter circuit. For this purpose a number of special stages were included in the circuit, including in each channel

phasometer through three rheostatic low-resistance amplifier stages with automatic control of their gain coefficient, without introducing any parasitic phase shift. The ARA system protects the main amplifier path of the instrument and the stages operating at comparatively large signal amplitudes, which makes it possible to substantially reduce the danger of amplitude-phase errors in the phasometric part of the fluorometer.

A tube voltmeter is built into the instrument, making it possible to monitor the levels of the light signals in both channels, which makes it possible to guarantee the absence of amplitude-phase errors in the photomultiplier tubes and the correct operation of the electronic part of the setup. Neutral filters are used to equalize the signals from the scatterer and from the substance under investigation. Owing to the design of the ARA system, the selection of the appropriate construction of the instrument, the selection of the most suitable photomultiplier-tube specimens, etc., the amplitude-phase characteristics of the instrument are such that a twofold change in the level of the light signals cannot lead to an error in measuring the value of \(\tau\) of more than \(10^{-10}\) sec. Therefore, in practice there is no need to establish equality of the signals with great accuracy, which facilitates the measurements.

The tube voltmeter also makes it possible to judge the presence of scattered light, the level of fluctuation noise, and the quality of the crossed light filters and thus to avoid errors associated with this. Let us note that improving the crossedness of the light filters is, as a rule, accompanied by the loss of part of the light (luminescence light, exciting light, or both). Therefore, when measuring the duration of weak luminescence, it may be preferable not to reduce an already weak signal, but to introduce into the measurement results corrections for imperfect crossedness. For this, in addition to the amplitude and phase of the signal from the cell filled with a solution of the substance under investigation (\(I_p\) and \(\varphi_p\)), it is necessary to know the signal amplitude when the cell is filled with pure solvent (\(I_0\)). Then, as follows from consideration of the vector diagram in Fig. 14, the phase angle of the luminescence light \(\varphi\), which determines its duration, can be found from the relation

Fig. 14. Vector diagram for considering the correction for imperfect crossedness of the light filters.

\[ \varphi=\varphi_p+\arcsin \frac{\sin\varphi_p} {\sqrt{1+\left(\frac{I_p}{I_0}\right)^2-2\left(\frac{I_p}{I_0}\right)\cos\varphi_p}} . \tag{10} \]

Correspondingly, additional corrections may be introduced for the level of the fluctuation voltage and for the luminescence of the solvent. Of course, the introduction of such corrections reduces the accuracy of the measurements.

The fluorometer described is designed for measurements by the null method with the introduction of a standard compensating phase shift. When the scatterer is placed in the measuring channel, the indicator is set to zero by means of special phase shifters provided in the electronic part, in order to compensate for random phase shifts in the setup itself. The scatterer is then replaced by a sample of the substance under investigation, and a neutral attenuator is first selected, as required for approximate equalization of the signal levels from the scatterer and the sample. The change then observed in the reading of the phase indicator is compensated by introducing the measured phase shift. For this purpose, the comparison channel of the instrument has an optical line 30 cm long, which makes it possible to introduce an opposing phase shift from \(0^\circ\) to \(4.3^\circ\) with an error of less than \(0.1^\circ\) (Fig. 10). In addition, the instrument has electrical step phase shifters, each step likewise being designed to introduce an opposing phase shift of \(4.3^\circ\) (Fig. 12). Depending on the value of \(\tau\), the quantity \(\varphi\) entering expression (9) may take values from \(0^\circ\) to \(90^\circ\). By combining the appropriate number of steps of the electrical phase shifters and varying the length of the line, this range can be covered smoothly, setting the required value of \(\varphi\) with an accuracy determined by the resolving power of the phase indicator. Such a system made it possible to use only a short optical line, which reduced the danger of errors associated with changes in the illumination conditions of the photocathode of the photoreceiver in the comparison channel. Finally, the presence of two types of phase shifters—optical and electrical—makes it possible to carry out their mutual checking systematically, which increases the reliability of the measurements.

5. CONCLUSION

The new phase fluorometer, developed as a result of a special investigation of the features of phase measurements and the identification of the principal sources of error in fluorometric studies, possesses a number of positive qualities. The resolving power of the constructed fluorometer at a low level of fluctuation noise is about \(2 \cdot 10^{-11}\) sec, which exceeds by at least an order of magnitude the resolving power of the best fluorometers described earlier (see Table 1). Correspondingly, the non-systematic errors in measuring the quantity \(\tau\), which, being greater than the resolving power of the instrument, determine its magnitude, also prove to be smaller. It is significant that the resolving power of the fluorometer remains high even at a relatively large level of fluctuation noise, which makes it possible to study the duration of fluorescence under weak luminescence.

To illustrate what has been said, we shall present some results of measurements on the constructed fluorometer.

The preservation of the comparatively high sensitivity of the fluorometer for weak signals made it possible to trace the concentration dependences of the luminescence duration of a number of substances. Studies of $\tau = f(c)$ for an alkaline solution of fluorescein showed that, when observing luminescence in thick layers, this dependence is nonmonotonic (Fig. 15). The character of the dependence established by us agrees with the results

Fig. 15. Graph of the concentration dependence of the duration of fluorescence of an alkaline solution of fluorescein: 1 and 5 — data obtained on a new fluorometer (respectively for a thick and a thin layer of solution); 2 — data of Bailey and Rollefson; 3 and 4 — Shmillen’s data for a thick layer and 6 — for a thin layer of solution.

of Shmillen^42 and Bailey and Rollefson^41. Comparison of these results with the course of the dependence $\tau = f(c)$ for thin layers of a fluorescein solution, and some other additional experiments carried out by us, indicate that the observed increase in the value of $\tau$ before the onset of concentration quenching is associated only with the effect of reabsorption. The numerical values of the change in $\tau$ show that reabsorption plays a more significant role than might have been expected on the basis of earlier works^14, ^39.

On the fluorometer that was constructed, the luminescence durations of several systems were also investigated: anthracene and its derivatives—polystyrene (Table III on p. 108).

The obtained values of $\tau$ agree well with ideas about the nature of luminophore—polystyrene systems and about the character of the interaction of anthracene and its derivatives with styrene during polymerization^48.

Table III

The high resolving power of the instrument makes it possible to study very short-lived processes with it. Among the substances we investigated, of some interest is the measurement of the duration of the luminescence of tetraphenylbutadiene in xylene, which has a value on the order of $8\cdot 10^{-10}$ sec, and is practically independent of concentration when the latter is changed by a factor of 20 (Fig. 16).

Fig. 16. Results of measuring the concentration dependence of the luminescence duration of tetraphenylbutadiene in xylene.

We were the first to measure the luminescence duration of dye centers in ionic crystals. It proved to be equal to several units of $10^{-9}$ sec, which agrees with P. P. Feofilov’s ideas about the electric dipole character of the luminescence of dye centers.

Experience with practical work on the new fluorometer gives reason to hope that with its aid it will be possible to carry out a more detailed and broader study of fluorescence decay times than

this has been done up to the present time. Analysis of the errors of phase measurements and identification of their sources may be useful in the construction of phase-measuring devices of the most varied purposes.

*
* *

The construction of the fluorometer described was begun at the suggestion of S. I. Vavilov in the laboratory which he headed. A large part in the work was taken by members of this laboratory—V. A. Molchanov and V. I. Shirokov, who put much labor into the development of the instrument.

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