864
J. LAVOREL
Miscellaneous Examples of the Transfer of Electronic Energy.-The transfer of energy of excitation is important in a number of spectroscopic and photochemical phenomena which are either too specialized or too complicated to be included in this discussion. Among these, the following cases deserve mention. In the polymers of dyes, such as pseudo isocyanine, rapid exciton migration leads to the appearance of a new narrow absorption and emission band.” Some of the complex phenomena observed by Moodie and Reid3* are probably the result of energy transfer in systems composed of acceptor molecules adsorbed on the surface of donor crystals or in mechanical mixtures of two different kinds of microcrystals. West d e m o n ~ t r a t e dthat ~ ~ naphthalene sensitizes the decomposition of ethyl iodide in hexane solutions. This reaction appears to involve collisions of the second kind between ethyl iodide molecules and naphthalene molecules in either their first excited singlet or lowest triplet state. It deserves more detailed study. The fluorescence of euwpium which is excited when solutions of europium salicylaldehyde, etc., are illuminated with light adsorbed only by the orcan be explained as due ganic part of the (37) For bibliography, see footnote 3, reference 32. (38) M. Moodie and C. Reid, J. Chhsm. Phye., 20, 1510 (1956). See also F. Weigert, Trans. Faraday Soc., 36, 1033 (1940). (39) ( a ) W. West and W. Miller, J . Chem. Phys., 8, 849 (1940); (b) W. West, Ann. N. Y . Acad. Sci., 41, 203 (1941). (40) ( a ) S. Weissman, J . Cham. Phya., 10, 214 (1942); (b) A. Sevchenko and A. Morachevskii, Izueat. Akad. Nauk Ser. Fiz., 15, 628
Vol. 61
to the inductive-resonance transfer of excitation from the organic part of the molecule to the europium. Bucher and Kaspers4’ observation that the carbon monoxide-myoglobin complex can be photochemically split as well by light which is absorbed by the protein as that which is absorbed by the prosthetic group, seems to require a similar explanabut even more convincing examt i ~ n .A~ similar ~ ple of energy transfer, is Bannister’s demonstrat i ~ that n ~ the ~ quantum yield of fluorescence of phycocyanin is the same when the exciting light is X 4050 A . which is absorbed by the chromophore or is X 2750 k. which is absorbed by the protein moiety of the molecule. Sensitized fluorescence occurs efficiently in lant cells.43 The fluorescence of chlorophyll-a is sensitized by chlorophyll-b and, in red algae and certain other plants, by other plant pigments. The transfer of excitation which is responsible for this sensitized fluorescence increases the efficiency of the utilization of solar energy by the plants. It has also been suggested5 that energy transfer by inductive resonance44 plays other and more important roles in the primary acts of the photosynthesis of green plants. (1951); (c) A. Sevchenko and A. Tr&mov, Zhur. Eksptl. Teoret. Fiz,, 21, 220 (1951). (41) (a) T. Bilcher and J. Kaspers, Biochim. Biophys. Acta, 1, 21 (1947); (b) T. Bilcher, Aduanc. Enzmol., 14, 32 (1953). (42) T. Bannister, Arch. Biochem. Biophys., 49, 222 (1954).
(43) E. Rabinowitch, ”Photosynthesis,” Vol. 11, Interscience Publishers, Inc., New York, N. Y., pp. 812-817 and 186&1879. (44) J. Franck, “Research in Photosynthesis,” Interscience Publishers, Inc., New York, N. Y.,1957.
EFFECT OF ENERGY MIGRATION ON FLUORESCENCE I N DYE SOLUTIONS1 BY J. LAVOREL~ Photosynthesis Laboratory at the Department of Botany of the University of Illinois, Urbana, Illinois Received February $1, 1067
The steady-state spatial distribution of excited molecules in an illuminated liquid medium exhibiting fluorescence has been calculated assuming the occurrence of energy migration by resonance transfers. Equations are derived for the effect of this phenomenon on the spectral distribution of fluorescence as it emerges from the vessel and on the observed action spectrum of fluorescence excitation, as well as for the analogous effect of secondary fluorescence; the possibility of distinguishing between the two effects is demonstrated. A tentative value of the mean displacement between absorption and emission due to resonance transfers is derived from experimental observations on alkaline fluorescein solutions; it is about 20 mfi in 5 x 10-2 molar solution, corresponding to a “random walk” over about 300 molecules.
1. Introduction The occurrence of electronic excitation energy migration in certain dye solutions has long been surmised from the observation of such phenomena as concentration-quenching and concentrationdepolarization of fluorescence.3 Unfortunately, when one seeks to derive from such observations quantitative conclusions as to the extent of energy migration, no easy way of achieving this aim ap-. pears. This is particularly true of self-quenching, (1) Work performed during the tenure of a Rockefeller Fellowship, with the assistance of the Office of Naval Research. (2) Laboratoire de Biologie physico-chimique, Facult6 des Sciences, Universit6 de Paris, Paris (France). (3) (a) F. Perrin, Compf. rend., 192, 1727 (1931); (b) J. Perrin. ‘2e Conseil d e Chim. Solvay,” Gauthier-Villara, Paris, 1925, p. 322.
where several different physical mechanismsbesides energy migration-may operate simultaneo~sly.~ I n what follows, we have looked a t this problem from a somewhat different side, starting with the basic idea that the first result of energy migration must be an increase in the distance between the place where absorption occurs and the place where the light is emitted, in excess of the displacement attributable to the diffusion of the primarily excited particle. Direct measurement of this displacement proved difficult; but analysis shows that this displacement must affect the intensity and spectral (4) Th. Forster, “Fluoresaenz Organischer Verbindungen,” Vandenhoeck & Ruprecht, 1951, p. 243.
.
July, 1957
EFFECT OF ENERGY MIGRATION ON FLUORESCENCE IN DYESOLUTIONS
distribution of the observed fluorescence and that this effect offers a promising approach to the quantitative study of energy migration. 2. Theory Effect of Energy Migration on the Spatial Distribution of Excited Molecules.-The steadystate distribution of excited molecules, which prevails when a dye solution in a rectangular vessel is illuminated with parallel light falling normally to the front wall of the vessel, is given by the BeerLambert law. It is easily shown that the observed intensity of “backward” fluorescence of wave length XI, -i.e., of the fluorescence emerging from the vessel in the direction opposite to that of the incident beamsarising from the elementary volume of unit cross section comprised between the planes z and z dx, is 2.1.
865
migrating energy when it reaches the front wall of the vessel. We can assume either that the energy is “absorbed” at the wall (which means n = 0 for z = 0), or that it is “reflected” by it (which means dn/dz = 0 for x = 0). These two cases will be designated “a” and “r,” respectively. One then finds for d F either (in case a)
(2.1.3A)
dF’
610
joc 1 - (jocdK)2 [e-jacx - j o c 2 / ~ ~ e - x / ~ ~ ] e - j ~(2.1.3B) c~ds
+
(if D = 0, both equations are reduced to the static case (2.1.1), as expected). The observed total fluorescence, F’, can be calculated easily from d F = e-j1cs+cU (2.1.1) (2.1.3). where I n the following, we will assume that the vessel is deep enough for the solution to absorb completely + = yield of fluorescence d l = absorbed intensity of the exciting beam (wave the exciting beam. Now, as DT = 1/6 ntd-2r, length = A,) or number of molecules excited per where nt is the frequency of transfers (which accordunit time in this elementary vol. , ~ d the j1 = absorption coefficient a t the wave length at which ing to Forster is proportional to c ~ ) and fluorescence is observed (XI) average intermolecular distance (which is proporc = concentration tional to c-’/a) and as r decreases when c increases, If it is assumed that between the absorption one can test equations (2.1.3A) and (2.1.3B) by and the emission, the excitation energy performs varying the concentration, c. However, energy a number of jumps from molecule to molecule, the migration is expected to cause only a small correcfluorescence quantum, produced by excitation a t tion in the over-all effect of c on F’, which will be x, will be emitted a t a somewhat different distance primarily due to changes in + caused by self-quenchA. Hence the observed ing. In order to eliminate this influence of $, we from the front wall, x fluorescence intensity will no longer be given by. proceeded as follows. (I) The intensity of fluorescence excited a t Xo (2.1.1). To estimate the effect of the displacements A on was observed a t two wave lengths-one, AI, where the observed fluorescence, the simplest hypothesis fluorescence is strongly reabsorbed ( j , >>O) and one can make is to regard energy migration as a one, XZ, where it is not reabsorbed at all ( j , = 0) “random walk’’ with elementary steps equal to the (cf. Fig 1); we called R(c) the ratio of the fluoresmean intermolecular distance, in other words, to cence intensities at these two wave lengths a t conassume that energy migration obeys the law of centration c. From F’ as given after integrating (2.1.3), one finds molecular diffusion. Then, if D is the “diffusion coefficient” of 1 R(c) = R(0) (case “aJJ) (2.1.4A) energy-and if one neglects the molecular diffu1 jlcdD> sion-the number, n, of excited molecules present in unit volume at x must obey, in the steady state, and the differential equation
+
+
(2.1.4B)
being the actual life time of excitation a t concentration c and jo the absorption coefficient for the wave length of excitation (intensity = 1 0 a t the front wall). This equation means that the production of excited molecules per unit time in a ,unit volume balances the loss of excited molecules by deactivation and by energy diffusion. T o solve equation (2.1.2), boundary conditions must be defined. Since the exciting beam is completely absorbed by the medium, the first boundary condition is n = 0 for x = 03. The second boundary condition concerns the behavior of T
( 5 ) Our formulaa are atrictly valid only if the solid angle of obaervation ia negligibly small-a condition which can hardly be met in actual experimentation. However, a correction for the finite angle of obaervation would not affect any of our conclusions.
where R(0) is the limiting value of R for very small c’s. (2) The fluorescence which is not re-absorbed (ie., that in the region X>Xf), was excited a t two wave lengths, Xo (at the maximum of the absorption curve), and ho’ (cf. Fig. 2). We called R’(c) the ratio of these two fluorescence intensities a t concentration c. As above, we get from (2.1.3) after integration R’(c) = R’(0) -Oc-l: + j dE (case “a”) 1 -I-~ ‘ o c ~ D T
(2.1.5A)
and R’(c) = R’(0) (6) Th. Forster, ref. 4, p. 85.
(case “r”)
(2.1.5B)
866
J. LAVOREL
- absorption, ---
fluorescence
Fig. 1.-Position of the wave lengths of excitation (ha) and observation (A1 and Xz) of fluorescence in the measurements of R . Notice strong re-absorption at X1.
- absorption ---
fluorescence.
Ab
Ao
Fig. 2.-Position of the wave lengths of excitation (A0 and XO’) and observation ( A > Af) in measurements of R’.
with R’(0) being the limiting value of the ratio R‘ for very small c’s. These relations hold true as long as the dependence of 4 upon X is not affected by variations of c. As a first approximation, we will assume this to be true. It may be noticed that the ratio R’(c) provides a better criterion to choose between hypotheses “a” and ‘Y’ than the ratio R(c),since R’(c) depends on c (more specifically, increases with c) only in case “a.” 2.2 Effect of Secondary Fluorescence on Observed Fluorescence Intensity.-So far, we have neglected to take into account secondary fluorescence, i e . , re-emission of reabsorbed primary fluorescence (which can occur in the wave length range where the absorption band and the fluorescence band overlap). This event can repeat itself until the energy reaches the boundary of the medium; it provides another “trivial” mechanism of energy migration. It must not be overlooked in the estimation of the extent of non-radiative energy migration due to resonance transfers. Because of absorption and re-emission, the observed fluorescence is the sum of the fluorescences of all possible orders m
m
F =
Fn = n=l
.fn[&(c)ln
(2.2.1)
n=l
where &(c) is a function decreasing from 1 to 0 with increasing c (because of self-quenching) , and where
VOl. 61
the fn’s depend on the wave lengths of excitation and observation, the amount of overlapping of the absorption and fluorescence bands, and the geometrical conditions of the experiment. The power series in &(c) originates in the fact that the fluorescence of order n escapes the medium after n absorption-and-emission acts, during which it has n occasions (in n different molecules) to disappear by self-quenching. Solving the problem of secondary fluorescence in its general form seemed to us a difficult undertaking; as the series (2.2.1) has to converge, we contented ourselves with the first two terms. (a) The first term follows readily from (2.1.2) by integration
J&
= loE(Ao)&(c)D(Xl)dX1
(2.2.2)
Comparison with (2.1.2) shows that we have now represented 4 as the product of three factors. (1) E(Xo)-the efficiency of excitation a t Xo. It assumes values from 1 to 0. (E(Xo)= 1 for the largest part of the absorption spectrum). (2) &(c) defined above. (3) D(X1)dXl--the intensity of fluorescence as function of the wave length of observation. When conditions are chosen such that E(h0) and Q(c) are both = 1, one has JD(X,) dX1 = quantum yield of fluorescence, the integration being performed over the fluorescence band. (b) To calculate the second term in (2.2.1), it is necessary to introduce a generalization of the Beer-Lambert law, to account for the fact that the primary fluorescence coming from a given elementary volume is propagated in all directions and not only in one direction as the primary exciting beam. Therefore, the absorption of primary fluorescence has to be treated by means of the Beer-Lambert law in spherically symmetrical form. This is accomplished if one puts ‘p
= poe-j(X)cp
(2.2.3)
instead of the usual I = Io e-j(X)cx,where (ao is the light flux from the elementary volume and cp the light flux through the spherical surface with the radius p , centered in this volume. For the elementary volume defined by an increment of e and p and revolution around the axis OX, as in Fig. 3, the amount of light absorbed which originated in the central point A, is pde2rrp sin 47rpz
e jcpOe-jO4v dp
(2.2.4)
For every part of this cylindrical volume, the amount of reabsorption suffered by the secondary fluorescence when traveling backward (in the negative X direction) is the same e-jdr
+
P COS
8)
(2.2.5)
*
cpo plays now the role of lo in (2.2.2), with the difference that this exciting light is no longer monochromatic. In fact, it consists of elementary contributions such as
TOE(Xo)Q(c)D(X)dXjoce-jocxdz
(2.2.6)
with X being restricted to the range of wave lengths shared by the absorption and the fluorescence band. We have also to take into account the yield
. I
July, 1957
EFFECT OF ENERGY MIGRATION ON FLUORESCENCE IN DYESOLUTIONS
867
of this secondary fluorescence, and the wave length a t which we observe it. This gives a factor (2.2.7)
E(h)Q(c)D(hi)dh
X being the same as in (2.2.6).
Combining (2.2.4), (2.2.5), (2.2.6) and (2.2.7), we get
The threefold integration over p and 2 corresponds to the infinite volume a t right of the interface P (Fig. 3) and the integration over h is carried forj(h), D(X) and E @ ) . The result is F Z = 51 I G E ( h o ) [ Q ( C ) I z D ( h ~ ) d h ~ 3 ~ Fig. 3.-Absorption of light flux originating in A in the elementary volume pde29p sin e.
We can now express the quantities R(c) and R’(c) come more and more important as concentration in terms of Fl and Fz (neglecting, for the time being, increases. Furthermore, the effects of the two diffusion phenomena on R(c) and R’(c) are opposite in any effect of energy migration). sign. Therefore, as c increases, R(c) must a t first (1) For R(c),we find increase because of secondary fluorescence, [cf. 1 K Z 1 KlQ(c) (2.2.10) R(c) = R(0) (2.2.10)1, and then decrease because of energy migra1 KI 1 Kz&(c) tion [cf. (2.1.4)];whileR’(c) mustfollow theopposite with trend, first decreasing as indicated by (2.2.11), and then increasing as suggested by (2.1.5A).
+ +
++
j(x) In (1 31
and KI =
5 py
In (1
+3
~ ) E(h)D(h) ] dh
+ &) + 11 E(h)D(h)dh
As K1 < Kt, it is seen that upon increasing c (thus decreasing &(c), from 1 t o O), R(c) must increase from R(o)to (1 Kz)/(l &) X R(0). (2) For R’(c),we find
+
Rye) = p ( 0 ) 1 1
+
Q @ + K’o x 11 +o Ko&(c) x
0
-
(2.2.11)
with
and K O=
JAp FIn (1 + 3&) + 1 1 E(h)D(h)dh
Here, K‘o > K O (as j ‘ o < jo), hence R’(c) must decrease from R’(o) to (1 &)/(l K’o) X R’(o) when c increases. 2.3. Combined Effects of Energy Migration and Secondary Fluorescence.-It often has been asked whether the effect of energy migration by resonance transfer could be distinguished from that of secondary fluore~cence.~The above calculations show that this is indeed possible. The intensity of secondary fluorescence should decrease, relatively to that of primary fluorescence, when c increases [cf. (2.2.1)]; its effect will therefore become less and less important with increased concentration. The frequency of resonance transfers, on the contrary, increases with c, so that its influence will be-
+
+
(7) A. Terenin, Uspekhi Fisich. Nauk, 68, 37 (1956).
3. Experimental Results and Discussion For the determination of R(c), the fluorescence of alkaline aqueous (1 M NaOH) solutions of fluorescein in a 6 mm. thick Lucite cuvette was excited by the 436 mp line isolated with filters (Wratten C5 No. 47 Farrand 422.13.28) from the spectrum of a low-pressure mercury vapor lamp (GE AH4, 100 w.). The image of the fluorescent volume was formed on the entrance slit of a Bausch and Lomb grating monochromator (focal length 250 mm., 600 grooves/mm.) which was usually operated with both slits set a t 1 mm. or less (dispersion 6.6 mp/mm.). The distance between the cuvette and the condensing lens was about 5 cm. The area of the lens was about 4 cm.2. The fluorescence intensity was measured by means of a photomultiplier (RCA No. 6217) clamped a t the exit slit of the monochromator. R’(c) was determined by exciting fluorescence with monochromatic light from the same monochromator and measured through a Corning filter (No. 3484) by means of the same photomultiplier. The distance between the cuvette and the entrance of the photomultiplier was about 10 cm. I n both cases, the optical axis of excitation and observation were practically perpendicular to the front wall of the cuvette, and the section of the exciting beam was much smaller than the area of the front wall of the cuvette. Figures 4 and 5 show the behavior of these two ratios. Figure 5 is of particular interest, for i t permits-as stated a t the end of section 2.1-to make a choice between the hypotheses “a” and “r.” The fact that the experimental curves of R’(c) as function of c go through a minimum, and increase again a t the high values of c, can only be
+
J. LAVOREL
A 1
10 -a
10 -2 C.
Fig. 4.-R(c)/R(O) as a function of c (mole/liter) for two values of hl.
Vol. 61
angle will be slightly larger for c2 than for c l ; furthermore, its variation with c will be the larger the smaller j~ (or j’o). This would cause R’(c) to increase with concentration. On the other hand, the extent of reabsorption of fluorescence within the solid angle of observation will increase upon its opening with increasing concentrations. This would cause R(c) to decrease. However, with the experimental set-up we have used, both effects were too small to affect significantly R(c) and R’(c). I n order to calculate 6, the square root of the mean square energy displacement between absorption and emission-which is a convenient measure of the extent of energy migration (6 is equal to 2 / 5 ) , on!! would need to know the shape of the curves R(c) and R’(c) as determined by secondary fluorescence alone; one could then ascribe the differences between these theoretical curves and the actually measured ones, found at high values of c, to energy migration. This procedure is unfortunately not possible, for (2.2.10) and (2.2.11) are simplified formulas, which do not take into account the fact that E(X) in (2.2.9) also depends on c, especially in the anti-Stokes region (observations which lead to this conclusion will be published later). We did, however, calculate values of 6 by comparing &’(or R) at a particular concentration with the minimum (or maximum) value of R’(or R ) , since these extreme observed values are likely to lie close to the limiting values of R’ or R as determined by secondary fluorescence alone. The results are listed in the following tables. TABLE I ( A ) 6 from R(c) measurements at c (6 = 3 mp) xo XI XI
=
(mr)
(mr)
(mr)
436 436
490 505
580 580
490 710
(2 = 2 mp)
10-2
10-3 C.
F i g . 5.--R’(c)/R’(O) as a function yf c (mole/liter) for four values of xo .
explained on the basis of hypothesis “a” (i.e., by assuming that the excitation energy is lost when it reaches the front wall). Another possible source of variation of R(c) and R’(c) is worth considering. Let us take two concentrations, c1 and c2 (c2 > el) and consider the solid angle of observation [cf. footnote 31 corresponding to two equivalent elements of the fluorescent volume (by “equivalent elements” we mean elements such as clzl = czz2). As xi > 2 2 , the solid
6
(mr)
(B) 6 from R’(c) measurements a t c
d
1.5 X 10+ d.I
xo
?.e’
(mr)
(md
490 440 490 450 490 475 480 490 E: mean intermolecular distance.
=
5 X lo-* Jf 6 (mr)
370 300 150 85
Of interest is the decrease of the 6 values when gets close to XO in Table IB. This suggests that assumption “a” is oversimplified, and that instead of speaking of an “energy absorbing interface,” one should better speak of an “energy absorbing houndary layer” of finite thickness. If this modification is accepted, the 6 values calculated from (2.1.4A) and (2.1.5A) become too large; however, the values a t the bottom of Table IB will be less affected by this error then those a t the top. As it is difficult to formulate quantitatively the hypothesis of the “absorbing layer” when solving eq. (2.1.2) (one would have to make D or 1/7-or both-functions of x), we have tried instead to extrapolate a value of 6 for jo’ tending toward j,. For this purpose, we write (2.1.5A) in the form
*
EFFECT OF ENERGY MIGRATION ON FLUORESCENCE IN DYE SOLUTIONS
July, 1957
869
where R‘ stands for R‘cc,/R’(o, and plot the first member of this equation as a function of (jo j’o) : the intercept of that curve with the axis of ordinates l/c.\/&, and hence 6 (Fig. 6). gives usj, I n this way, we found 6 % 20.5 mp. (This average displacement by energy diffusion is appreciably larger than the average displacement, Amol, due to molecular diffusion. Applying Einstein8formula to the case in point, we find Am01 = 0.7 m@. Hence, neglect of molecular diffusion in the derivation of eq. (2.1.2) was justified.) With a r-value of about S ~ C . ,the ~ average time which the energy spends in a single molecule becomes l/nt S 3X sec., and the number of visits during the life time of excitation nt7 = 300 (for c = 5 X
+
10-2 M ) .
Forster’s purely theoretical formula for the relative probability of energy transfer applied to fluorescein a t the same concentration,’O leads to somewhat less “optimistic” values: l/nt % 18 X 10-’2 sec., and ntr E 50. 4. Conclusion One must note that the proposed method implies ascribing all changes, which the action spectrum of fluorescence excitation, the absorption and fluorescence bands exhibit upon increase in concentration, to secondary fluorescence and energy migration, i.e., assuming the absence of any true changes in the shape of the latter. Offhand, no such changes would be expected to occur-leaving aside the known cases of dimerization or polymerization-but we cannot be quite sure that they are non-existent”; and this possibility constitutes the (8) A. Einstein, 2. EEeklrochem., 14, 235 (1908). (9) 9. W. Cram, 2.Phvsik, 109, 551 (1936). (10) Th. Forster, ibid., 103, 177 (1936). (11) After the manuscript of this paper had been submitted t o the editor, continuation of this study produced evidence of a concentration dependence of the absorption spectrum of fluorescein (see also Soderborg,lP Ghosh and Sengupta”), which can be interpreted as due to the superposition of two dimer bands upon the monomer band. However, the dimerization of fluorescein is much weaker than t h a t of many other dyes, such as thionin or methylene blue. At the highest concentration used (5 X lo-* M ) the ratio [monomer]: [dimer] still is of the order of 4. The non-fluoresrent character of the dimer could be quantitatively proved by parallel examination of the concentration dependence of the absorption and the fluorescence excitation efficiency, E @ ) , (cf. our remark on E(X) in Section 3). A non-phenomenological interpreta. tion of E ( M and Cl(c) has been made possible by these findings, and is being worked on.
50
100 150 ( j , jo’) x 10-8. Fig. 6.-Extrapolating procedure for 6. The i n t e r c e p t , the l/cdD~). curve with the axis of ordinates = log ( j ,
-
+
main limitation of the reliability of our method. Nevertheless, we believe that our calculations rest on a sound basis-namely, on visualizing energy migration as a diffusion process. We cannot claim to have obtained anything better than a first, crude approximation of the true value of the diffusion reach 6, but the very fact that such an evaluation proved possible, in other words, that the behavior of R and R’ as function of c proved to be in qualitative agreement with theoretical calculations, provides a new argument in support of the hypothesis that resonance energy migration does take place in concentrated solutions of fluorescent dyes. My thanks are due to Dr. E. Rabinowitch, as well as to Dr. R. Emerson and the other members of the Photosynthesis Laboratory a t the University of Illinois for their kind assistance in making this work possible. The change in the absorption spectrum should be taken into account in the interpretation of the ratios R and R’. We found, however, t h a t the ratio R(c)-at least for XI =490 mr-is not significantly afratio on which we fected by dimerieation. As for R’(c)-the relied in the calculation of the migration path I-estimate8 show t h a t correction for the absorption by the dimer will not modify essentially the conclusions drawn from this ratio in the present paper: specifically, the “r” hypothesis remains incompatible with the exp’erimental data, and the calculated &values still range from 30 to 150 m r . (12) B. Soderborg, Ann. Phvsik, 41, 381 (1913). (13) J. C. Ghosh and S. B. Sengupta, Z. p h y e i k . Chem., B41, 117 (1938).
