THE J O U R N A L O F
PHYSICAL CHEMISTRY Regi:itered i n U.S.Patent Office 0 Copyright, 1979, by the American Chemical Society
VOLUME 83, NUMBER 5
MARCH 8,1979
This paper is largely in the nature of a review article, though some novel developments are included. It is intended by the authors and the editors to be of service to the community of spectroscopists and photochemists who wish to use the polarization techniques discussed, and the focus is on the photochemical and spectroscopic utility rather than on the devices and techniques themselves. We hope the paper will indeed serve this purpose, and we would welcome comments from readers as to the desirability of similar articles.
Applications of the Photoelastic Modulator to Polarization Spectroscopy' K. W. Hlpps and G. A. Crosby" Chemical Physics Program, Washington State University, Pullman, Washington 99 164 (Received June 13, 1978; Revised Manuscript Received September 25, 1978) Publication costs assisted by the Air Force Office of Scientifk Research, Directorate of Chemical Sciences
A review and discussion of the properties of the photoelastic modulator pertinent to polarization spectrclscopy is presented. Detailed analyses of' linear and circular polarization spectrometers for both absorption and emission are given. To demonstrate the results obtainable with these spectrometers, the magnetically induced ciircular polarization of emission from tris(4,7-diphenyl-l,10-phenanthroline)ruthenium(II) chloride and the polarization ratio spectrum of the emission from tris(2,2'-bipyridine)ruthenium(II) chloride are presented.
Introduction The generation of light of fixed polarization has been known for centuries, and the application of polarization techniques to spectroscopy is as old as spectroscopy itself.2 Recently, however, a renewed effort to interpret theoretically the interaction of polarized light with matter has developed, complemented by experimental advances in the design of polarization apparata that rely on modern dynamic methods of measurement. Our purpose is to review both the static and dynamic methods of producing and analyzing palarized light and then to focus on the newest of the devices employed for dynamic polarization, the photoelastic modulator (PEM). Since virtually all polarization spectroscopies can be performed with ease and high sensitivity using the PEM, we describe its applications to a variety of problems. For illustration, examples of data taken on spectrometers utilizing this device are included. Static Methods. Linear polarization (LP) may be produced by a variety of devices, which generally fall into two classes, dichroic and nondichroic. The dichroic polarizer relies on the preferential absorption of light of a given polarization by a material. Such devices have very large angular apertures, some frequency dependence of 0022-3654/79/2083-0555$0 1 .OO/O
polarization efficiency, and a strong frequency dependence of transmission. They are not useful for high power light beams. A familiar example of the dichroic polarizer is commercial Polaroid. Nondichroic polarizers are of two basic types, those utilizing reflection and those employing refraction. When a beam of light, not at normal incidence, is reflected from a dielectric surface, it is generally partially polarized parallel to that surface. When the incidence angle is the Brewster angle, the reflected light is totally polarized; the transmitted light is only partially polarized. Thle reflection-type polarizer is relatively achromatic and can be constructed for any wavelength region for which a transparent dielectric is available. The geometric restrictions inherent in this device, however, make it the least attractive choice for wavelength regions where eiither dichroic or refractive-type polarizers are available. Refractive polarizers take advantage of the difference in path and/or indices of refraction of orthogonally polarized beams in uniaxial crystals. Either one of the beams is reflected into an absorbing material or both are transmitted but physically separated. These polarizers are usually achromatic over a large wavelength region, can 0 1979 American Chemical Society
556
K. W. Hipps and G. A. Crosby
The Journal of Physical Chemistry, Vol. 83,No. 5, 1979
handle high powers, and have the smallest intensity losses of all types. In principle, polarizers should make excellent analyzers. That they are, at best, inconvenient for spectroscopic applications is a consequence of the characteristics of available detection equipment. The efficiencies of both monochromators and photomultiplier tubes are polarization dependent. This polarization sensitivity is also wavelength dependent. Thus, for quantitative measurements, a detailed calibration curve is required. Another disadvantage appears when the net linear polarization is low. Small errors in the intensities observed for orthogonal polarizations can lead to gross errors in the net polarization spectrum. A fundamental difficulty with all polarizers is that they must be physically rotated to determine the polarization state. Circular polarization cannot be analyzed with linear polarizers. The general technique for analyzing circularly polarized light (CPL) is to phase shift it to linearly polarized light and then to analyze the latter for direction of polarization. Conversely, the same optical elements placed in inverse order can convert an unpolarized beam to an elliptically polarized one. There are several static devices in common use for the production of CPL, and we list them with their spectroscopically pertinent properties. (1) Fresnel Rhomb. An achromatic device with high transmission efficiency. Its principal disadvantages are a small angular aperture and physical displacement of the emergent beam. (2) Quarter Wave Plate. A precision cut slice of a uniaxial crystal; it is useful only over a small wavelength region and has a small (55') angular aperture. (3) Variable Quarter Wave Plate. An example is the Babinet-Solei1 compensator. It has all the features of the quarter wave plate except that it can be manually adjusted for operation a t various wavelengths. For a "wavelength-at-a-time'' experiment, this is an excellent static device. Because a polarizer must be used with all of these devices, they possess all the difficulties mentioned earlier in connection with the measurement of linearly polarized light. In summary, static devices for measuring and producing polarized light are troublesome to work with and offer low sensitivities where the net polarization is also low. Dynamic Methods. In dynamic methods the polarization state is varied periodically in time with a period much longer than one oscillation of the electric field. The signal is then resolved either by Fourier analysis or by phase sensitive detection. For example, consider a device utilizing a rotating polarizer3 to analyze-the polarization direction of a beam with electric field E: E = E,(COS x l + sin ~.7) (1) Let this beam pass through a polarizer rotating at frequency w and then impinge on a photomultiplier. With the pass direction of the analyzer as fi fi = cos (ut);+ sin (cot); (2) we calculate the transmitted intensity, I , to be I = lE't.ZI2 = (I0/2)[1 + cos (2wt - 2x)l
(3)
We see that the zero frequency (dc) component is one half the original intensity and that x appears as a phase shift of the harmonic at 20. The advantage of this device lies in its convenience. For optical rotation measurements where the light is monochromatic before entering the sample, we are now a t liberty to scan the optical rotation spectrum [x(X)], provided that the scan time is much
greater than l / w . For a full appreciation of this device, Scholtens et al. should be c o n ~ u l t e d . ~ Rather than rotating optical elements mechanically, one can utilize the reduction in symmetry caused by external fields to produce dynamic polarization. Piezoelectricity is a phenomenon demonstrated by certain crystals whereby a stress generates an electric fielda4 The converse piezoelectric effect is the generation of strain due to an applied electric field. Since strain affects the refractive indices of a crystal, electronically driven piezoelectric crystals can be used to modulate the polarization state of a transmitted beam. These devices that depend on the piezoelectric effect are called Pockels cells. Their advantages and disadvantages have been discussed in the l i t e r a t ~ r e .Their ~ ~ ~ main features are (a) small angular apertures, typically about 2', (b) the requirement of high voltages (kV) and powers, and (c) the need for a piezoelectric crystal that is transparent in the spectral range of interest. The disadvantages of the Pockels cell may be circumvented by the use of the photoelastic modulator. This device was not invented until 1966.'i8 It has since been used for ellipsometry? visible circular dichroism (CD),9 IRCD,lOJ1linear dichroism (LD),l1J2 circularly polarized emission (CPE),I3 magnetically induced circular polarization of emission (MCE),14and polarized Raman studies.16 Application to MCD has been so widespread that the reader is urged to consult the references in Stephens.16 The basic concept is the application of a periodic stress to an isotropic material, thereby inducing a periodic variation in the optic ellipsoid, at a frequency that is a resonant frequency for that material. Since the material is isotropic in the absence of stress, the angular aperture is about an order of magnitude larger than for the Pockels cell.5 Since all materials show photoelasticity, a modulator may be constructed for virtually any wavelength range. Because the device functions at resonance, very low input voltages (- 20 V) and powers are required. These devices generally operate in the 200-20-kHz region, although other frequencies can be ~ s e d . ~ i ~ , ~ J ~ Various methods have been used to construct a PEM;5,7,8J7 the one described and operationally analyzed below is a commercial device of the type discussed by Kemp5 and manufactured by Morvue Electronics Corporation. A single crystal quartz bar is employed as a piezoelectric transducer to drive a fused quartz bar, and the crystal and fused quartz bars are matched such that a node occurs in the strain at their junction. T h e Photoelastic Modulator Linearly Polarized Incident Radiation. In this section we present a detailed quantitative analysis of the effect of the modulator and an analyzer on the time dependence and polarization state of a partially linearly polarized beam. These results will be used in the analyses of the various spectrometers described later. First, we quantify the physical nature of the PEM (Figure 1). A 50-kHz electric field is applied to a -18' X-cut single crystal of quartz.8 Glued to it is an identically shaped block of fused quartz, the geometry chosen to give a fundamental acoustic vibration of wavelength A, = 21. The extensional displacements are along j and nearly independent of x and z. If we take the origin to be the center of the face (x,y system shown), the displacement, u , along the bar is given by u = uo sin (2ny/X,) cos (w,t) (4) where uo is a linear function of the applied maximum voltage (V,) and W , is the fundamental angular frequency.
The Journal of Physical Chemistry, Vol. 83, No. 5, 1979 557
Applications of the Photoelastic Modulator Detector
Figure 1. A diagram of the photoelastic modulator including the specification of coordinates used in the mathematical analysis of its operation.
The strain, t, however, is what determines the refractive index variation in the bar: c
au
3
-=
aY
UoT
- cos 1
(?)
cos (w,t)
(5)
We see that the strain, and therefore the refractive index, varies sinusoidally and has a broad maximum at the center of the fused quartz. For the amplitudes used in most applications, it is found experimentally that the phase shift, I$ 1 (n, - nY)w29/X,is a linear function of t. We write $4 =
$0
GVm cos (umt)= -cos (u,t) x
sin aj)
(7)
After passing through the modulator a t time t , the 3 component has suffered a relative phase shift d,(t) = d,o cos (a&),to produce the field
12,"
=
cos a; +
el$
sin a:)
(8)
2," represents elliptically
polarized light whose polarization state varies in time. If $o is taken to be 9, at wmt = -r/2!we have linearly polarized light, at w,t =: -913 we have elliptically polarized light, and a t u,t = 0 the beam is again linearly polarized but a t 90" to its original direction. When an analyzer whose passage direction is d (line AA in Figure 1 ) where a' = ( l / f i ) G +
7)
(9)
is inserted, the detector will see only one polarization at all times and wavelengths, but the detected intensity, Id", will oscillate with time according to the expression Ida
- = Io" == la'.Em12 -[I
+ sin (2a)cos $1
2
(10)
cos 14Jo cos (u,t)I = Jo(d,o) + m
k=l
Ida
IO*
= -([1
2
+ sin (2a)J0(d,0)] + m
2 sin (2a)C (-l)kJzk(40) cos (2kw,t)) ( 1 2 )
If we define &"(nu,) as the amplitude of the signal of frequency nu,, we have IO"
+ sin ( 2 0 l ) J ~ ( d , ~ ) ]
Ida(0) = - [ 1 2
(13)
and
I,ia(2kw,) = (-1)k11,"J2k(d,o) sin (2a)
(14)
Note that, except for d,o = 0, there will always be slome component of intensity at 2wm as long as a # n a / 2 . For a given even harmonic and fixed values of a and I; the amplitude will depend on JZk(d,o).In order to maxiimize the amplitudes of a given harmonic, d,o should be set (via V,) to make JZk($o) a maximum. For practical applications, the second harmonic is generally the highest harmonic used. Figure 2 shows Jo,J1,and J2;J1is included for future reference. Suppoge there is a second, incoherent, polarized component, E@, in the beam whose polarization angle is a n / 2 (Figure 1).
+
E@= Eo@(-sina i + cos ay)
105)
Its intensity a t the detector is IdB
IO@
= --{[I
-. sin
(2a)J0(d,0)] m
2 sin (2a) (-l)kJzk(d,o)cos (2kumt)) (16)
where I,3ais the intensity of the original a-polarized beam. In order to demonstrate the time dependence of' Id', one expands cos d, in a Fourier series18 2
where Jl(d,o)is a Bessels function of integer order. This allows eq 10 to be written as
k=l
(6)
where d,o = GV,/X and G is independent of both the incident light wavelength, A, and V,. d,o is the maximum relative phase shift induced in a light beam during a period of the bar's oscillation. Consider a single polarization state, $", of the incident beam:
i 3 = E ~ ( C O S a;+
Flgure 2. A plot of the Bessel's functions Jo, J,, and J2 as a function of do.
(-1)kJ2k(d,o)cos (2hwnt) (11)
k=l
The total intensity seen by the detector is the sum of Ida and I d p and is given by
K. W. Hipps and G. A. Crosby
The Journal of Physical Chemistry, Vol. 83, No. 5, 1979
558
To appreciate the significance of this result, note that is linear in the difference in intensity between a and p polarizations and that the detection system always sees a single polarization. Thus, both of the difficulties associated with static devices have been removed. Further, since $o is a function of V,/X, V , may be kept proportional to X so that I d depends only on the intrinsic wavelength dependence of the incident beam. For convenience we define Id(2w,)
I = Io* + l o o and AI = Io* - loo
PEM Drive
1 - 1
I I
(18)
and eq 17a and 17b become
I-----j(01
Id(2ko,) = AI sin (2a)(-l)kJ2k($O)
(19b)
In later sections we will show the relationship of these results to physical measurements. Circularly Polarized Incident Light. Imagine a pure right circularly polarized beam (Figure 1)with electric field
where $+ is the basis vector for pure right circularly polarized light. After passage through the modulator the electric field is
As for the linear case, the polarization is now modulated. With the analyzer removed, the intensity seen by the detector is independent of time and equal to Io+. If we replace the analyzer, it selects out the a‘ component to give = lti.Z’,+12 = (1~+/2)[1- sin
Id+
Expansion of
Id+
$1
(22)
in a Fourier series yields m
Id+
= (10+/2){1
+ 2C(-1)kf1Jzk+l($o) k=O
cos [(2k + l)wmt]] (23)
Since third and higher harmonics are seldom used, we neglect them in all that follows. From eq 23 it is clear that the dc term is independent of $o and equal to half the original beam intensity. Except for the dc component, no even harmonics contribute to Id+. This circumstance makes it possible to analyze linear and circular contributions simultaneously. If the light to be analyzed is composed of both left and right circularly polarized beams with independent phases, a superposition of their intensities at the detector will be seen. Writing Io = Io+ + Io(24) and we have and Id(wm)
= AIJl(($O)
(26b)
As in eq 19, the difference in polarization amplitudes is directly accessible, and the detection system need not be
v(kw,)
Flgure 3. Schematic diagram of tne detection system used for polarization studies.
calibrated for polarization effects. The difficulties inherent in static measurements of differential circular polarization have been avoided.
Physical Arrangements for Various Spectroscopies Preliminaries. In the previous sections we have provided the basic equations and definitions that underlie spectrometer designs. We now proceed to relate these quantities to experimentally measurable parameters. The most direct method of determining the Fourier coefficients of an electrical signal is to use a tuned amplifier. For our purposes, a lock-in amplifier that can be frequency and phase matched to 0, or its second harmonic is appropriate. Most lock-in amplifiers display a signal that is proportional to the root mean square voltage (D) applied. The dc component of the signal is usually measured by a dc current meter. We must convert equations such as (19) and (26) from intensity equations to voltage and current expressions. Further, because detection and conversion machinery affect the values of voltage and current measured for a given intensity, we must account for the properties of the devices as the equations are transformed. Figure 3 is a block diagram of a generalized intensity analyzing system. The photoelectric transducer (PT) will usually be a photomultiplier, although photoconductive and photovoltaic devices have been used for work in the infrared.lOJ1 The current from the PT, i(t), depends on several factors that are included in eq 27. G’(p,X) is the i(W,X) = Id(w,X)G ’(P,X)F(Id)D(W) (27) gain of the PT in units of current/energy at wavelength h for polarization state p of Id(t). For our applications, p will usually be fixed and G’(p,X) = G,,’(M (28) F ( I d ) is a correction factor due to possible nonlinearity in the detector. It may usually be set equal to unity by an appropriate choice of detector and voltage range. D(w,z) represents the electrical attenuation of a signal of frequency w by the impedance (z) of the phototube and load resistor (R) circuit. Since most photodetectors have an internal capacitance on the order of picofarads, this is a real problem. For an EM1 9558 PT, a choice of R 1 20 kB can lead to serious degradation of a 50-kHz signal. Since the choice of PT effectively leaves only R as a variable, we set D(w,z) = D(w,R) (29) and eq 27 becomes i(w,X) = Id(w,X)G,’(X)D(w,R) = Id(w,X)Go(w,X) (30)
The Journal of Physical Chemistry, Vol. 83, No.
Applications of the Photoelastic Modulator
the emission is focused on the sample by a lens. Because strong 2w, signals can affect the w, reading of the lock-in, the source and X selector should be chosen so that the exciting light is unpolarized. If this is not possible, linearly polarized light polarized parallel to a modulator axis should be used. The CP emission enters the modulator head and then passes through a polarizer, P. The polarizer is oriented as in Figure 1. The intensity at the monochromator is now specified by eq 26a and 26b. Because the monochromator has both wavelength and polarization sensitivity, the intensity at the detector becomes I&O) = (Io/2)Gp”(X) (354
(Exdtatlonf Source
a Selector
tens Sample Modulator
5, 7979 559
a Ramp Amp1 1 fie r
Polarizer ( P ) Monochromator Control
Io(wm) = AIJ1(40)Gp”(X)
(35b)
Detector
iOM
V(i)
Figure 4. Schematic diagram of a spectrometer for measuring polarized emission.
where G,”(X) specifies the gain of the monochromator for (fixed) polarization p . Using previous results, we have io(M Gp(X)D(0,R)(Io/2) (36a) and
where Go(w,X)depends parametrically on the fixed values of p and R. The lock-in amplifier senses the signal u(w,X) = Ri(w,X). The use of a differential amplifier input assists the removal of the dc portion of the signal. This signal is then mixed with a reference signal at the frequency to be analyzed (kw,), and the rms value of u(kw,,h) is available either as a meter reading or a dc voltage proportional to ij(kw,,X). Because some losses of high frequency signal occur in the leads and differential amplifier, d(ko,,X) is given by ij(w,X) = Rd(w)i(w,X) (31) where d ( o ) is usually -1 and may be incorporated into Go(w,X). We then have b(w,h) = RG0(w,X)Id(w,X)/fi
ij(kw,,X) = RGo(kw,,X)~d(kw,,X)/fi
(334 (33b)
In what follows, the components enclosed by dashes in Figure 3 will be denoted simply as the detector. The outpute will be governed by the equations given above. If Oliois expressed in units of mV/pA, D’(kw,) 10-3R/d2. Emission. Circular Polarization. Optically active molecules will emit light that is preferentially circularly polarized. In the presence of a magnetic field virtually all materials should demonstrate preferential circular po1ari~ation.l~ In both cases a measurement of AI = I- - I+ as a furiction of the emission wavelength, A, should give nonzero results. The naturally circularly polarized emission spectrum (CPE) is AI(X), and the normalized CPE is A I / I . The magnetically induced circularly polarized emission (MCE) spectrum is AI&); the normalized MCE is AIHIIH. Except for the presence of an external magnetic field for the latter measurement, both CPE and MCIS can be recorded by the same device. An emission circular polarization spectrometer is depicted in Figure 4. Exciting light containing no wavelength components overlapping
-
-
AI( A) = d( h)K1/ GP(A)
(37b)
and (37c)
(32)
where I ( w , X ) = I(w,h)/d/Zhas been used. Since we are interested only in frequencies that are multiples of w,, all the above results may be collected as eq 33a and 33b or 34 where D’(kw,) is ~ R / d 2
i(0,X) io(X)= Go(O,X)Id(O,X)
where Gp(X)= G,’(X)G,”(X). As shown in Figure 4,we couple the modulator voltage, V,, to the wavelength drive so that 4ois constant. A glance at Figure 2 shows that the sensitivity R i maximum at $o x / 2 . Since R is fixed for the experiment, we may write lo(X) = Koio(X)/Gp(X) (374
where
Equation 3721-c allow us to extract the total emission. CP emission, and normalized CPE spectra from i j ( X ) and io(X). The constants K1 and KOare best determined experimentally. In practice, since absolute emission intensities are seldom reported, the ratio K l / K ois all that is needed. This may be obtained in several ways. One method involves producing pure circularly polarized light a t several wavelengths via a Babinet-Solei1 compensator. This method gave a value of 0.15 pA/mV when R was 10 kQ and q50 = ~ / for 2 our instrument; eq 38 predicts 0.122 pA/mV. There are a few points that deserve special attention. (1)The measured sign of the d signal depends on the lock-in phase setting and the polarity of the modulator drive signal. Before and after every experimental run the sign and magnitude of K,/Ko should be determined for at least one wavelength to ensure correct operation of the spectrometer. ( 2 ) Because of the broad maximum of Jl(40)about #o = 1.7, a very high precision coupling of the monochromaitor to the modulator is not necessary. (3) If filtering of the emitted light is desirable, filters should be placed after the polarizer P. This will affect G(X) but not the normalized CPE. The polarization distortion introduced by putting anything between the emission source and modulator is usually high. Due to the sensitive nature of this spectrometer, even the finest quality strain
500
The Journal of Physical Chemistry, Vol. 83, No. 5, 1979
free material can produce spurious signals unless its surfaces are parallel to the modulator surface. (4) Because the experiments are carried out with the modulator and analyzer in a fixed orientation, G,(h) need only be determined for that orientation. The signal-to-noise ratio of this type of spectrometer in the visible and UV regions is theoretically limited by the fact that discrete photons are counted. An analysis by Gafni and Steinberg13 shows that S / N N r/q% (39) where r is the fractional circular polarization, q is the quantum efficiency of the detector, N is the average number of photons per second incident on the detector, and 7 is the time constant used on the lock-in amplifier. S/N values within a factor of 2 of eq 39 have been achieved with our system. In the IR, thermal radiation from room temperature surfaces is an important source of noise and stray radiation. In this region it is prerequisite that the source be chopped to discriminate against background radiation. Emission. Linear Polarization. The properties of interest in the linearly polarized emission (LPE) experiment are the unpolarized emission intensity, IN, and the difference in linearly polarized intensities parallel and perpendicular to some axis, AZ = I,,- I,. For natural LPE, the axes of interest are crystallographically fixed. In magnetically induced LPE (MLPE),parallel refers to the field direction. In either case, eq 19a and 19b are appropriate for the intensity at the monochromator. A glance at these equations shows that a considerable simplification of analysis ensues when the following conditions obtain: (1)cry_tal cor fielcl) parallel axis set at +45" to modulator axis (E" = E45= Ellin Figure 1);(2) cjo set at 2.405 so that Jo($o)= 0. This causes little loss in 20, sensitivity since J2(2.4)/J2(3.4) = 0.92 (Figure 2). When the above restrictions are imposed, eq 19a and 19b become &(O) = P/2 (404 and Id(20m)
= -MJ2(cjO)
(40b)
Again taking the wavelength and frequency dependences of monochromator and detector into account, we have in analogy to eq 37a-c I = Koio(X)/G,(M (414 A I = Ls(X)K~/G,(X)
(41b)
where 1.03
K2/Ko
- &RJ2(2.4)
PA
mV
(42)
This choice of $o removes all polarization sensitivity from the dc signal. Under these conditions, the normalized LPE is now given simply as (43) Although it is not directly apparent, eq 41 and 43 depend critically on the angle of the analyzer with respect to the modulator axis. It must be exactly 45". By careful adjustment of this angle we have achieved the measurement of a constant I within 0.5% while an incident purely linearly polarized beam was rotated through 360". Variations of I within 5% are easily attained and should be more than sufficient for recording emissions that are less than 20% linearly polarized. Our experimental value of
-
K. W. Hipps and G.A. Crosby
K 2 / K o 0.286 kA/mV with a 10 K value for R compares to 0.165 pA/mV from eq 42; the error was due principally to the fact that 100 kHz (20~)was near the high frequency cutoff for the electronics used. In light of the above, Figure 4 also represents a LPE spectrometer provided only that conditions 1 and 2 are met. It is interesting to note that J1(2.4)/J1(7r/2) N 0.9. A t little cost in sensitivity the spectrometer in Figure 4 can be permanently set for doing both LPE and CPE; the only adjustments required are the phase and frequency of the lock-in amplifier. Caution: When performing each experiment care must be taken to ensure that linearly polarized light from the excitation source is not reaching the detector. Further, polarization selection problems must also be eliminated.19*20 Polarization Ratio Spectroscopy. We first gave a brief account of this spectroscopy that usually goes by the unfortunate name of luminescence polarization spectroscopy and is sometimes called polarization selection spectroscopy. Consider an isotropic matrix in which tiny dipoles are randomly oriented. When excited with linearly polarized light, those dipoles that have their axes parallel to the incident polarization vector will be selectively excited. Subsequent reemission will be preferentially polarized in the same direction as the original excitation. If the relative polarization P is defined as
I,,- I , (44) a detailed statistical analysis gives P = +1/2 for this ideal case. Alternatively, a more complex model for a molecule may be used. If absorption and emission directions are at 90" to each other in the molecular coordinate system, the emission will be preferentially polarized perpendicular to the incident polarization and will give a negative value for P. For the general case see the detailed analysis in Feofilov.2 The experimental method is to excite the sample at various wavelengths and determine P at each wavelength. Alternatively, the emission may be scanned for fixed wavelength (polarized) excitation. This gives information about the symmetries of the absorbing and emitting states. Our application of the modulator to this experiment may be the first of its kind. It is clear from the discussion of linear polarization how the experiment should be performed. The only modification required is that the X selector should be a monochromator with a polarizer affixed. The pass direction of this polarizer should be at 45' to the modulator axes. If the emission is observed at 90" rather than at 180" as in Figure 5, then less stringent requirements are placed on the filtering system. We point out that the same sort of experiments may be done with circularly polarized light.19,20The spectrometer depicted in Figure 4 can also be used for this purpose. Linear and Circular Dichroism. The spectrometer of Figure 4 may be used for polarized absorption measurements as well, although there are two difficulties associated with this design. Truly unpolarized light must be used for excitation, and extreme measures are required to avoid spurious signals caused by scattered light. Both these problems can be avoided by the design shown in Figure 5. Irradiation of the sample with modulated light removes the first problem, and the use of a double monochromator eliminates the second. We are left only with the requirement that the detector be somewhat insensitive to polarization state. Note first that the polarizer P is oriented at 45' to the modulator axes so that the field transmitted by the
The Jormal of Physical Chemistry, Vol. 83, No. 5, 1979 561
Applications of the Photoelastic Modulator
modulator is given by eq 8 with
Eo E , = -(i +
fi
+
(Y
I
=~/4:
+
Source
1
(45) Monochromator
or, in a circular basis
+ Polar
A?-' Modulator
Modulator
In the CD case, the sample transmits the electric field
ip
E o ( ]+ i e @ ) e - t - c d P ~ ~- i &() e - f1 +cd/Z+ =: #++2 -J- (47) 2 where t* are the extinction coefficients for right and left circullarly polarized light. The resulting intensity is e
-e+cd
+
e-c-cd)
+ 10 - sin
4 ( p + c d - e-s-ed)
(48)
2 2 By the techniques used earlier one obtains b(w,)/io = K tanh [(Aecd)/2]
Control
Detector
jy
= 12 (
I
i(h)
Figure 5. Schematic diagram of a spectrometer for measuring polarized absorption.
n
(49)
where At 1 (6- - t+). For Atcd C 0.15, eq 49 may be replaced by O l i o = AcKcd/2 (50) to better than 1% . For linear dichroism the analysis is slightly more complicated due to the fact that the sample x and y axes are at 45" to the modulator x and y axes. Denoting the samplle directions (1 and Iwith extinction coefficients t I and respectively, we expand eq 45 in the basis Zll a d e', given by
550
600
650
700 nm
Figure 6. Tris(4,7-diphenyl- 1,1O-phenanthroline)ruthenium(II) clhloride in poly(methy1methacrylate) at -4.2 K: ---, emission spectrum (I); -, magnetically induced circularly polarized emission (AI) at !50 kG. For illustration A l h a s been multiplied by a factor of 18.
for LD, The relations Atc = t- - E+ and AeL = ell - tl have been used, and K and K' are determined experimentally. Therefore ,@,Ia
= (E0/2)[(1
+ e@)e'Il+ (1 -
]
(52)
and the intensity transmitted by the sample is IL
I=
( I o / 2 ) [ ( e - W+ e-f-l-cd)
+ (e-4cd
- e-flcd
cos 41 (53)
Use of eq 11. allows us to write eq 53 as IL(0)= ( 1 0 / 2 ) [ ( e - f ~ ~ ced- f l c d ) (e-fllcd- e-f,cd)Jo(40)] (544 and
+
P~(2w,)=
+
+IoJz(40)[e-"-~d
- e-ellcd 1
(54b)
The 2wm component will be proportional to At = el - t i for small values of AEcd but will not be simply re\ated otherwise. Further, oliowill vary as tanh (Atcd/2) only when J o ( ~ ois )zero. In summary, we have given the design features for a spectrometer that can measure either linear or circular dichroism. By the novel technique of maintaining $o at 2.405 rather than the a / 2 or P, as is customary in the literature, both the LD and CD may be recorded. The experimental parameter AEcd is given by (55)
for CD, and by
Discussion In order to exemplify the results obtainable from a PEM polarization spectrometer, we present an emission polarization ratio and a MCE spectrum obtained from our construction of the spectrometer of Figure 4.21 The excitation source was a 1000-W mercury-xenon lamp filtered to provide excitation energy only in the wavelength region of the first intense charge-transfer band (xy polarized)22 of the D3 complexes studied. The molecules of interest were dispersed in a poly(methy1 methacrylate) matrix and housed in the bore of an Oxford Instruments cryostat with concentric superconducting solenoid. The commercially produced modulator was followed by a 30" angular aperture Glan-Thompson polarizer. Due to the dijffuse nature of the emission, a Spex Minimate with 1-2-nm band pass was used as the analyzing monochromator. For the polarization ratio experiments, a Glan-Thompson polarizer was inserted before the sample. Figure 6 displays the actual emission ( I ) and MCE ( A I ) data obtained for tris(4,7-diphenyl-l,lO-phenanthroline)ruthenium(II) chloride at -4.2 K and 50 kG. The maximum in AZ represents only 5% of the total emission intensity a t this wavelength. It should be noted that the MCE shows considerably more structure than the noirmal emission. Figure 7 presents the intensity and relative polarization of emission (I -l 11) l from tris(2,2'-bipyridine)ruthenium(II) chloride at 77 K. The value of P obtained was roughly constant at +0.15 across the b,and. The data reported in Figures 6 and 7 are uncorrected data, exactly as taken from the recorder. Obvious improvements on our spectrometer would be the use of electronic ratiometric recording, extended red response detectors, and
562
The Journal of Physical Chemistry, Vol. 83, No. 5, 1979
\,
fP\
\
\
I I
i
\
I
I
I I
\
I
\
I I / L
A ' 550
I
I
I
600
I
650
700
750 nm
Flgure 7.
Tris(2,2'-bipyridine)ruthenium(IJ) chloride in poly(methy1 methacrylate) at 77 K; ---, emission spectrum ( I ) ;-, differential linear polarization (AI = 111- II). I n the figure, AI has been multiplied by a factor of 7.4.
the insertion of a tuned preamplifier before the lock-in.
References and Notes (1) Research supported by AFOSR(NC) USAF Grants 72-2207 and 76-2932. (2) P. P. Feofilov, "The physical Basis of PolarizedEmission", Consultants Bureau, New York, 1961, contains a thorough review of polarization as applied to spectroscopy before 1960. (3) D. J. Scholtens, J. F. Kleibeuker, and J. Kommandeur, Rev. Sc/. Instrum., 44, 153 (1973).
J. N. White and W. C. Gardiner (4) J. F. Nye, "Physical Properties of Crystals", Oxford University Press, London, 1969. (5) J. C. Kemp, J . Opt. SOC.Am., 59, 950 (1969). (6) L. Velluz, M. Legrand, and M. Grosjean, "Optical Circular Dichroism", Academic Press, New York, 1965; B. C . Cavenett and 0. Sowersby, J . Phys. E, 8, 365 (1975). (7) M. Biliardon and J. Badoz, C. R. Acad. Sci., Paris, Ser. 8 , 263, 139 (1966). (8)'s. N. JaSDerSOn and S.E. Schnatterlv. Rev. Sci. Instrum.. 40. 761 (1969). J. C. Southerland, G. D. Cimino, and J. T. Lowe, Rev. Sci. Instrum., 47, 356 (1975); 0.Schnepp, S.Allen, and E. F. Pearson, ibid., 41, 1136 (1970); B. H. Breeze and Bacon Ke, Anal. Biochem., 50, 281 (1972); W. C. Johnson, Jr., Rev. Sci. Instrum., 42, 1283 (1971); J. C. Coliingwood, P. Day, R. G. Denning, P. N. Quested, and T. R. Snellgrove, J . Phys. E, 7, 991 (1974). G. A. Osborne, J. C. Cheng, and P. J. Stephens, Rev. Sci. Instrum., 44, 10 (1973); J. C. Cheng, L. A. Naffie, and P. J. Stephens, J . Opt. SOC.Am., 65, 1031 (1975); R. Clark and P. J. Stephens, Roc. SPIE, 112, 127 (1977). I. Chabay and G. Holzwarth, Appl. Opt., 14, 454 (1975). R. Gale, A. J. McCaffery, and R. Shatwell, Chem. Phys. Lett., 17, 416 (1972). A. Gafni and I. Z. Steinberg, Photochem. Photobiol., 15, 93 (1972). R. Shatweil and A. J. McCaffery, J . Phys. E , 7,297 (1974); Mol. fhys., 30, 1489 (1975); H. G. Brittain, F. S. Richardson, J. P. Jasinski, W. C. Yeakel, and P. N. Schatz, J . Phys. Chem., 80,2228 (1976); K. W. Hipps and G. A. Crosby, Opt. Eng., 17, 176 (1978). L. I. Horvath and A. J. McCaffery, J. Chem. SOC.,faraday Trans. 2,562 (1977). P. J. Stephens, Annu. Rev. fhys. Chem., 25, 201 (1974). G. Baldacchini and L. F. Mollenauer, Appl. Opt., 11, 2676 (1972). H. Abramowitz and I. Stegun, "Handbook of Mathematical Functions AMS 55", National Bureau of Standards, Washington, D.C., 1965. K. W. Hipps, Chem. Phys., 23, 451 (1977); J . Phys. Chem., 02, 609 (1978). C. A. Emeis and L. J. Oosterhoff, Chem. fhys. Lett., 1, 129 (1967) For a discussion of the data, see ref 14. R. A. Palmer and T. S. Piper, Inorg. Chem., 5, 864 (1966).
An Evaluation of Methane Combustion Mechanisms. 2. Comparison of Model Predictions with Experimental Data from Shock-Initiated Combustion of C2H2,CzH4,and C2H6 J. N. White and W. C. Gardiner, Jr.* Department of Chemistry, University of Texas, Austin, Texas 78712 (Received June 19, 1978; Revlsed Manuscript Received November 3, 1978) fubllcation costs assisted by The Robert A. Welch Foundation and the Petroleum Research Fund
In earlier modeling studies on methane ignition it was established that satisfactory agreement between model predictions and methane combustion experiments can only be achieved when extensive account is taken of elementary reactions forming and destroying C2 hydrocarbons. To provide further tests of the completeness and accuracy of an optimized methane ignition model we simulated the shock-initiated combustion of the Cz hydrocarbons for a variety of conditions typical of experiments reported in the literature, determining the same diagnositics of reaction progress in the simulations as were used in the experimental studies. Overall the agreement is not nearly as good as was the case for studies on CH4 combustion. It would appear that the literature data base is internally inconsistent, and that some rather significant parts of the reaction mechanism are also still missing.
Introduction In recent years the quantity and quality of the experimental data on hydrocarbon combustion have reached the point where it has become possible and reasonable to develop detailed reaction mechanisms to account for the observations. In most explicit form, these mechanisms comprise large, putatively complete sets of elementary reactions, for most of which independent rate constant 0022-3654/79/2083-0562$0 1.OO/O
parameters are available; those elementary reactions not accessible to direct rate studies are assigned rate constants either on the basis of simple theories of rate processes (an assignment that is particularly appropriate for elementary reactions whose rate constants do not critically influence the course of reaction) or by comparison of combustion data with model predictions. Since CH4 combustion would appear to be the one that should have the simplest reaction 0 1979 American
Chemical Soclety
