J. Phys. Chem. 1984, 88. 3199-3203 experiment, the time course of the color change of the probe was monitored at X = 420 nm and reported as a photograph of an M picric acid and 2 X oscilloscope trace.lg When 2 X M surfactant solutions were mixed at 25 OC,two relaxations were observed, with kl = 7;’ = 36 s-l and k2 = 71’< 2.8 s-l determined by the use of the peeling-off procedure. Using the published relaxation curve, we added a spike (with k, = 1.00 s-’ and of comparable amplitude) to the discretized data and then applied the Z-transform method. By the use of only 35 equally spaced data points, we were able to recover the time constant of the spike as k , = 1.04 sw1 and to determine kl = 3 1.6 f 1 . 3 s-’ and k2 = 2.9 f 0.16 s-’. Clearly, these values should be viewed as more accurate than the ones determined by peeling-off, since the latter is only an approximation method. The second example represents an analysis of data collected on positron annihilation in Teflon at room temperatureS2O The positron lifetime spectra were measured with a standard timing spectrometer2’utilizing 5 cm x 5 cm plastic scintillators optically coupled to RCA 8575 photomultiplier tubes. The instrument could resolve lifetimes to 100 ps, with the time-to-pulse height converter calibrated to less than 1% standard error. The source of positrons of about 40 pCi was prepared by using 22NaC1in a sandwich arrangement between two pieces of -4.8 mg/cm2 gold foils. The data analyzed with the by-now standard POSITRON FIT^^ program resulted in three exponentials with time constants of k l = 3.17 ns-I, k2 = 0.886 ns-I, and k3 = 0.236 ns-I. As a comparison, by using 41 1 data points of the decaying portion of the curve, with the Z transform the time constants were computed as k l = 3.09 f 0.17 ns-’, k2 = 0.934 f 0.14 ns-I, and k3 = 0.211 f 0.022 ns-l. The precision of all the reported values is given as mean deviation from the mean. (17) K. Tamura and Z. A. Schelly, J. Phys. Chem., 84, 2996 (1980). (18) T. Svedberg and K. L. Pederson, “The Ultracentrifuge”, Oxford University Press, Oxford, 1940. (19) K. Tamura and Z. A. Schelly, J . Am. Chem. Soc., 101,7643 (1979). (20) S. D. Hyatt, thesis, to be submitted at UTA. (21) S. C. Sharma and S. D. Hyatt, to be submitted for publication. (22) P. Kirkegaard and M. Eldrup, Comp. Phys. Commun.,3,240 (1972).
3199
Conclusions The Z transformation is specifically tailored for the analysis of functions with a set of equally spaced discrete data. Since it involves no numerical integration, it has a definite advantage over other methods which do. Its inherent automatic smoothing, through eq 7, is very effective in the case of small noise. If, however, the S/N ratio is less than 20, additional smoothing by the method of zero determinants’O is recommended, prior to the application of the computational algorithm described. The quality of the results increases with the number of channels M used. According to our experience, for satisfactory results, M should be large enough for MD > 3 / k , , where k , is the time constant of the slowest component of the decay curve. Clearly, if the amplitude of one of the exponentials is smaller than the others by about a factor of ten, or if the time constants are somewhat less separated than required, the parameters of the decay components cannot be recovered to an acceptable degree of accuracy. Nevertheless, even in such unfavorable situations, the spike recovery method introduced yields the correct number of exponentials n in the decay curve, which is of major importance in the relaxation kinetic investigation of complex chemical system~.~~ Naturally, depending on the phenomenon described by eq 2, t may represent any independent variable other than time. In sedimentation equilibrium in the ultracentrifuge,I8 for example, distance square r2,and a parameter related to the buoyant molecular weight W, are the quantities corresponding to t , and ki, respectively. The simplicity of the algorithm, and our demonstrated resolution of three exponentials with less separation between the time constants than required for other approaches, makes the method of Z transform with spike recovery to an excellent candidate for application in many areas of the physical and biological sciences.
Acknowledgment. This work was partially supported by the R. A. Welch Foundation and the Organized Research Fund of UTA. We thank Dr. S. C. Sharma for supplying the positron annihilation data.
Spatial Distribution of Paramagnetic Entities in y-Irradiated 2-Methyltetrahydrofuran Glass As Studied by the Electron Spin Echo Method Tsuneki Ichikawa* and Hiroshi Yoshida Faculty of Engineering, Hokkaido University, Kita- ku, Sapporo 060, Japan (Received: November 9, 1983)
In order to examine the paramagnetic relaxation and the spatial distribution of radiation-generated paramagnetic entities, electron spin echo measurements were made for biphenyl anions (Bp-e) and localized electrons (elm-)in glassy 2-methyltetrahydrofuran (MTHF) matrices with and without solute biphenyl irradiated with y-rays at 77 K and at various radiation doses. The rates of both the longitudinal and transverse relaxations were found to increase with the increase in radiation dose, though the relaxation kinetics did not follow a simple exponential behavior because of various magnetic environments for Bp-. and elm-. Comparison with the results for chemically generated and randomly distributed Bp-- in the glassy MTHF matrix revealed that the paramagnetic relaxations of the radiation-generated Bp-- and elm-are largely due to the magnetic dipolar interaction with the free radicals formed from MTHF molecules. The average distance between Bp-. and the paired free radical originating from a single ionization event was estimated to be 4.4 f 0.4 nm from the relaxation rate extrapolated to zero dose.
Introduction Either of the primary chemical steps in the effect of high-energy radiation on molecular substances, ionization or bond dissociation, necessarily leads to the pairwise formation of paramagnetic entities, Le., a radical cation and an excess electron (or a secondarily formed radical anion) or a pair of neutral free radicals. In rigid matrices the spatial correlation resulting from the pairwise formation will 0022-3654/84/2088-3199$01.50/0
be more or less retained when these entities are trapped and immobilized. The painvise formation of the entities, together with the formation of spurs, blobs, and tracks,’ results in an inhomogeneous spatial distribution of reactive transients, which has long (1) A. Mozumer, “Advances in Radiation Chemistry”, Vol. 1, M . Burton and J. L. Magee, Eds., Wiley-Interscience, New York, 1969, p 1.
0 1984 American Chemical Society
3200 The Journal of Physical Chemistry, Vol. 88, No. 15, 1984
been of paramount interest in radiation chemistry. Direct proving of the inhomogeneous distribution of the reactive, paramagnetic entities trapped in rigid matrices has so far been made almost exclusively by examining their paramagnetic relaxation by means of the ESR power-saturation method. This method has been applied to localized electrons in glassy matrices at low temperatures and to2-5 free radicals trapped in irradiated and in frozen hydrocarbon^.^ Briefly it has led to the conclusion that the local concentration derived from the ESR power-saturation method is generally higher than the bulk concentration, which indicates the inhomogeneous distribution of the trapped entities. However, some uncertainties are still involved in the results obtained by this method, because the analysis of observed saturation curves is not straightforward and their implication is complex. It has sometimes been assumed without appropriate reasoning that only the transverse relaxation time, T2, depends on the distance between the paramagnetic entities, but the longitudinal relaxation time, TI, does not.2333s39 It is desirable to study the paramagnetic relaxation and the spatial distribution of the radiation-generated entities by a more direct method. The electron spin echo (ESE) method provides more unambiguous data on paramagnetic relaxation.1° It has been used by Tsvetkov et al. to study the spatial inhomogeneity of CHzOH radicals in solid methanol'l and 0- anion in aqueous alkaline glass'* irradiated with ionizing radiation of different LET. It has been shown that the local concentration of these paramagnetic entities is higher for the high LET radiation because of the track formation. However, the spatial correlation between a pair of the entities generated by a single ionization event caused by low LET radiation has not yet been elucidated. The present paper will report an ESE study on the spatial distribution of paramagnetic entities trapped in y-irradiated glassy 2-methyltetrahydrofuran (MTHF) matrices. The ESE measurements have been made on localized electrons in neat M T H F and on biphenyl radical anions in MTHF containing biphenyl as electron scavenger. The glassy M T H F matrix is one of the most extensively studied organic rigid matrices, where the excess electrons are localized and immobilized efficiently. The molecular cation of MTHF, the counterpart of an excess electron, is transformed into a more stable free radical giving a seven-line ESR s p e c t r ~ m ~through ~ , ' ~ the reaction with an intact M T H F molecule in close vicinity:
(2) D. R. Smith and J. J. Pieroni, Can. J . Chem., 43, 876 (1965). (3) D. P. Lin and L. Kevan, J . Chem. Phys., 55, 2629 (1971). (4) L. Kevan, 'Advances in Radiation Chemistry", Vol. 4, M. Burton and J. L. Magee, Eds., Wiley-Interscience,New York, 1974, p 181, and literature cited therein. 15) H. Hase. I. Hiromitsu. and T. Hiaashimura, J. Phys. Chem., 28, 17 ( 1982). (6) A. T. Bullock and L. H. Sutcliffe, Trans. Faraday Soc., 60, 2112 (1964). (7) H. Yoshida, K. Hayashi, and S.Okamura, Ark. Kemi, 23, 177 (1964). (8) S.Shimada, Y. Hori, and H. Kashiwabara, Radiat. Phys. Chem., 19, 33 (1982). (9) K. Toriyama, H. Muto, K. Nunome, M. Fukaya, and M. Iwasaki, Radiat. Phys. Chem., 18, 1041 (1981). (10) K.M. Salikov and Yu. D. Tsvetkov, "Time Domain Electron Spin Resonance", L. Kevan and R. N. Schwartz, Eds., Wiley-Interscience, New York, 1979, p 231. (11) A. M. Raitsimring, Yu. D. Tsvetkov, and V. M. Moralev, Int. J. Radiat. Phys. Chem., 5, 249 (1973). (12) Reference 10, p 268. (13) F. S. Dainton and G. A. Salmon, Proc. R . Soc. London, Ser. A , 285, 319 (1965). (14) C. Chachaty, A. Forchioni, J. Desalos, and M. Aris, Int. J . Radiat. Phys. Chem., 2, 69 (1970).
Ichikawa and Yoshida
DoselMrad Figure 1. Yield of (A)localized electrons in neat MTHF, (0)biphenyl anions in M T H F containing 0.06 mol/dm3 biphenyl, and (A)free radicals from M T H F molecules giving a seven-line ESR spectrum (both with and without biphenyl in MTHF) as a function of the radiation dose.
It has been alternatively proposed that the seven-line spectrum is due to another type of radical originating from the M T H F molecular cation.15J6 Nevertheless, the spatial correlation is expected between a localized electron and a free radical giving the seven-line ESR spectrum. In the presence of biphenyl, the localized electron is converted into the biphenyl radical anion,17 so that the radical anion will have a similar spatial correlation with the free radical. Experimental Section Reagent-grade M T H F was washed with aqueous NaOH solution (5 mol/dm3), distilled over Na metal, vacuum-distilled over Na-K alloy, and degassed by the freezing-pumping-thawing procedure. Three kinds of samples were prepared as follows: (1) neat MTHF, (2) MTHF containing 0.06 mol/dm3 biphenyl, and (3) M T H F solution of biphenyl radical anions generated by reducing biphenyl with N a metal. These samples were sealed under vacuum into quartz ESR tubes of 5-mm outer diameter and frozen into the glassy state by cooling them in liquid nitrogen. Samples 1 and 2 were irradiated at 77 K with 6oCoy-rays at a dose rate of 0.45 Mrd/h under complete darkness. The concentrations of paramagnetic entities in the samples were determined by comparing the intensities of their ESR spectra with the intensity of the ESR spectrum of DPPH of a known concentration. All the spectra were recorded at 77 K with a Varian E-109 X-band spectrometer. ESE measurements were made at 77 K with a home-built X-band spectrometer, which generated the microwave pulse of 50-11swidth and 1.O-kW maximum power. The measurements were actually made with 50-W microwave power incident into the cavity (rectangular TElo2, loaded Q of about 500). The transverse relaxation was examined from the dependence of the intensity of the 90°-180° spin echo on the time interval between the first and the second pulse, T . The longitudinal relaxation was examined from the dependence of the two-pulse spin echo signal intensity at a fixed 7,600 ns, on the repetition period, t, of the two-pulse echo series. Results Formation of Paramagnetic Entities by y-Radiolysis. Neat glassy M T H F irradiated a t 77 K shows the well-known ESR spectral curveI3-l5 composed of a sharp single-line spectrum of 0.45-mT peak-to-peak width due to localized electrons (elw-) and a seven-line spectrum of 2.0-mT average hyperfine separation due to the free radicals (R-) already mentioned in the Introduction. The yield of Re increases proportionally with the radiation dose as can be seen in Figure 1. The slope of the proportional increase gives a G value (number of entities generated per 100-eV energy absorbed) of 3.4. The elm- yield readily reaches a plateau value, (15) D. R. Smith and J. J. Pieroni, Can. J . Chem., 43, 2141 (1970). (16) T. Ichikawa, H. Yoshida, and K. Hayashi, J . Nucl. Sci. Technol., 9, 538 (1972). (17) J. R. Miller, J . Phys. Chem., 82, 767 (1978).
The Journal of Physical Chemistry, Vol. 88, No. 15, 1984 3201
Paramagnetic Entities in y-Irradiated 2-MTHF
ct
L
/
/
90.5
c TIPS
Figure 2. Examples of two-pulse echo envelope decays for (A) 1.4 mmol/dm3 dl,, (B) 1.4 mmol/dm3 Bp-- generated by y-radiolysis in glassy M T H F matrices at 77 K, and (C) 1.2 mmol/dm' Bp-.-Na+ generated chemically in M T H F solvent and frozen at 77 K. Measurements were made at 77 K with the time interval T between the two microwave pulses.
but the G value can be determined to be 2.9 from the initial slope of the dose-yield curve (see also Figure 1). This value is in agreement with those reported p r e v i ~ u s l y . ' ~ - ' ~ Glassy M T H F with 0.6 mol/dm3 biphenyl shows the superposition of the seven-line spectrum due to R. and a comparatively narrow spectrum of about 3.0-mT total width with poorly resolved hyperfine structure. The latter spectrum is due to biphenyl radical anions (Bp-.), The yield of Bp-. increases also proportionally with the radiation dose as can be seen in Figure 1, The G value for the Bp-- formation is 2.9. The G values of Bp-- and R- close to each other seem to indicate the pairwise formation of Re and the excess electron and the almost complete conversion of the latter into Bp--. Transverse Relaxation. The ESE was observed at 77 K for elm-and Bp-- generated by y-rays by fixing the external magnetic field at the center of their ESR spectra. Figure 2 shows examples of the decays of the echo intensity (echo envelopes) as a function of T . The decay curves are well expressed by V(27) = Vmd(7) exp(-aT
- mT2)
(2)
where Vmd represents the nuclear modulation effect arising from anisotropic hyperfine coupling with nearby proton^.'^*'^ The exponential factor denotes the transverse relaxation (phasememory decay). The value of m determined by analyzing the observed decay curves lay in the range (0.3-0.4) X 1OI2 s - ~for all the concentrations of both elm-and Bp-.. The average value was 0.36 X 1OI2 s - ~ . N o systematic trend was found in the variation of m. Therefore, it is concluded that m = (0.36 f 0.4) X 1OI2 s-* and is independent of the kinds of paramagnetic entities and their concentration. The transverse relaxation given by exp(-m$) arises from the interaction between electron spins and nuclear spins of surrounding protons." Figure 3 shows the dependence of a relaxation rate constant a on the concentration of el,-or Bp-.. The dependence is expressed as
+ 0.164[Bp-a]) X lo6 s-l a = (0.40 + 0.123[elm-]) X lo6 s-l
a = (0.40
(3) (4)
where [Bp-a] and [e,,-] are the concentrations of Bp-. and el,given in mmol/dm3. These empirical relationships were determined by the analysis of the decay curves by using the fixed value of m = 0.36 X 1OI2s2.It was found that the decay curves were independent of the incident power of microwave pulses (and therefore the width of the pulses). This and the linear dependence of a on the concentration indicate that the relaxation given by (18) W. B. Mims, Phys. Rev. B, 5, 2409 (1972). (19) L. Kevan, "Time Domain Electron Spin Resonance", L. Kevan and R. N . Schwartz, Eds., Wiley-Interscience, New York, 1979, p 279.
OO
I
I
I
5
10
15
Conc./mmoldm3 Figure 3. Concentration-dependent rate constant, a, of the transverse relaxation for (A)el=- and (0)Bp-. in y-irradiated M T H F matrices and ( 0 ) for frozen solution of Bp-S-Na' in MTHF. The two-pulse echo envelope decays are expressed by the functional form of exp(-as - m?). See also the text.
"0
1.0 ( t /ms)Q"
20
Figure 4.
Examples of observed recovery kinetics of longitudinal magnetization monitored by the two-pulse spin echoes with repetition period of t for (A) 1.4 mmol/dm3 elm- and (0) 1.4 mmol/dm3 Bp-. in y-irradiated M T H F matrices and ( 0 ) for frozen solution of 1.2 mmol/dm3 Bp-.-Na+ in MTHF. Measurements at 7 7 K.
exp(-m) is caused by spectral diffusion. The observed spin echo was reduced in intensity by a factor of more than 10 when either elm- or Bp-. was eliminated by photobleaching. This fact indicates that the overlapping of the spin echo due to R. can be neglected, so that the observed decay curves are regarded as being due solely to either el,- or Bp-.. The magnetic interaction between R. and elm-or Bp-. will affect the transverse relaxation of the latter. In order to check this, we examined the paramagnetic relaxation of biphenyl radical anions produced by chemical reduction of biphenyl with Na metal. These radical anions are free from R.. They will be designated as Bp--Na+ hereinafter, although they were found, from the absence of the nuclear modulation due to 23Na(see Figure 2), not to be associated closely with Na+ ions. The transverse relaxation of Bp---Na+ can be expressed by eq 2 with the common value of m = 0.36 X 10l2 s - ~and a = (0.24
+ 0.088[Bp-.-Na+])
X lo6 s-l
(5)
Although the a value is also linearly dependent on the concentration, it is generally smaller than that for B g - , so that the relaxation is slower. This difference will be attributed to the interaction between Bp-- and R- present in the y-irradiated samples. Longitudinal Relaxation. The dependence of the spin echo intensity on the repetition time t was examined for e,, Bp-., and Bp--Na+. The dependence shows the recovery kinetics of magnetization to its thermal equilibrium value after the saturation
3202 The Journal of Physical Chemistry, Vol. 88, No. 15, 1984
firstly under an assumption that R. radicals are distributed randomly around elo; or Bp-.. Let us denote the paramagnetic spins acting as donors as A spins and the spins acting as acceptors as B spins. The former in the present case are elm- or Bp-. which are excited by the microwave pulses. The latter are the spins under off-resonance conditions and are mainly Re. The relaxation rate of A spins due to the dipolar interactions with B spins is given by20
DoselMrad
’
2
8
Ichikawa and Yoshida
where ri is the distance from a particular A spin to the i-th B spin, and Oiis the angle between the vector pointing from the A spin to the B spin and the external magnetic field. If both T I and T2 of B spins are long enough, as is the case for R.,f(Oi)approaches $p? l m m l dm3 Figure 5. Dependence of the rate constant, b, of the longitudinal relaxation at 77 K on the concentration for (A) elm-,(0) Bp-., and ( 0 ) Bp-.-Na+. The observed kinetics of the relaxation are expressed by the functional form of V ( t ) / V o= 1 - exp(-bt0,82).
pulse. As can be seen in Figure 4, the recovery kinetics follow the equation
V(t) = Vo(l - exp(-bt0,82))
for Bp-a
+ 3.88[Bp-.]) X ( l o 3 s-1)o,82 + 1.21) x (103 S - y 2
(7)
+ 0.881) X (lo3 s-1)o,82
(8)
for elo; b = (0.61 for Bp---Na+ b = (0.14
+ 0.036[Bp-.-Na+])
X
(lo3 s-1)o,82
k = CArF6(1 - 3 cos2 Oi)2 i
(6)
for all the paramagnetic entities and all the concentrations examined. Scatter of observed results gives the uncertainty of f0.02 of the exponent of t . The rate constant, b, of the longitudinal relaxation increases linearly with the concentration for Bp; and Bp--Na+, as is shown in Figure 5. For radiation-generated Bp--, the proportionality holds between the radiation dose and the concentration. Therefore, b is linearly dependent on the radiation dose, I. For eloc-,the proportionality does not hold between Z and the concentration. In this case, b depends linearly not on the concentration but on I , as can be seen in Figure 5 . These results are expressed by the following empirical relationships:
b = (0.67 = (0.67
where Au is the difference in resonance frequency between the A and B spins. In such a situation, the relaxation is caused solely by the cross-relaxation due to dipolar interactions, Le., flip-flop between the A and B spins, and the relaxation rate constant can be written as
(9)
where I is expressed in megarads and all the concentrations are in mmol/dm3. It is evident that the longitudinal relaxation is much faster in the irradiated samples. This suggests strongly that the excitation transfer from elW-or Bp-. to R. opens an efficient pathway for the relaxation. This view is supported by the fact that the rate constant b for elW-is dependent linearly on the concentration of Re which increases in concentration linearly with the radiation dose.
Discussion Assumption of Randomly Distributed R.. The present results indicate that both the transverse and longitudinal relaxations depend on the concentration of the paramagnetic entities. However, only the rate constant, b, for the longitudinal relaxation will hereinafter be discussed in terms of the spatial distribution in the irradiated samples, because the experimental results are less scattered and their analysis is more straightforward for the longitudinal relaxation than for the transverse relaxation. The longitudinal relaxation will be analyzed semiquantitatively by the excitation transfer mechanism due to magnetic dipolar interaction
(12)
by using a factor A independent of ri and Oi. The observed recovery kinetics of the magnetization of A spins cannot simply be expressed by k, because each A spin sees a different spatial distribution of B spins around itself. The recovery kinetics deviates from the normal exponential behavior. Under the assumption of a random distribution of B spins, we have obtained the following expression for the recovery kinetics by averaging exp(-CAr;6( 1 - 3 cos2 for all possible locations of B spins:
( [
V(t) = Vo 1 - exp - (16T)T1’2C(At)1/2]) (9)(31’2)
(13)
where C is the number density of B spins. The recovery kinetics should follow an exponential function of and the rate constant is proportional with the concentration of B spins. If the spin diffusion within A spins becomes dominant compared with the cross-relaxation between the A and B spin systems, every A spin will see the same averaged distribution of B spins. This leads to the deviation of the exponent o f t from to unity. The present results gives the value of 0.82 for the exponent o f t . It is very likely that the rate of flip-flop between A spins is comparable to that of the flip-flop between A and B spins in the present cases. The rate constant b for Bp-. was found to be much larger than that for Bp-.-Na+. Since biphenyl anions are the only paramagnetic entities in the Bp-.-Na+ samples, the empirical relationship 9 gives the rate constant for the relaxation of biphenyl radical anions embedded in the MTHF glassy matrix free from other paramagnetic entities. By attributing the large difference in b between Bp-. and Bp-.-Na+ to the effect of R. radicals acting as B spins, the empirical relationship 7 for Bp-. is separated into two contributions, one from the “intrinsic” relaxation of the biphenyl anions and the other from the cross-relaxation with Re, by using the dose-yield curve shown in Figure 1: b= ((0.14 + 0.036[Bp-.]) + (0.53 + 0.307[R.])) X (lo3 s-1)o,82 (14)
The rate constant for the cross-relaxation with R. (the second term) is linearly dependent on the R. concentration, as is suggested by eq 13. The relaxation path of 0.53 X (lo3 s-1)o,82is due to the “residual” concentration of R., which is attributed to the count(20) A. Abragam, “The Principle of Nuclear Magnetism”, Oxford University Press, London, 1961, Chapter 8.
The Journal of Physical Chemistry, Vol. 88, No. 15, 1984 3203
Paramagnetic Entities in y-Irradiated 2-MTHF
1.5 -
1.0h
v L
.e-
r-0
r4 Y
I/
I
I
-5
0 In ( A ri6t
$ 0.5 -
5
Figure 6. Numerically computed recovery kinetics of the longitudinal magnetization of Bp-. based on eq 17 derived under the assumption that the relaxation is due solely to the excitation transfer to a counterpart R.. The dashed line shows the most plausible approximation of the computed curve by a straight line, V(t) = V, (1 - e~p[-fr,,-~(At)’/*]].Hatched area
shows the region that V(r)/V,= 0.1-0.9 a n d f = 0.94-1.14. erpart R. radical generated in pair with the Bp-a radical anion. The pairwise formation of R. has been ignored in the present assumption of the random distribution of R-. Relationship 14 gives an estimate that the contribution from the counterpart R. radical to the relaxation of Bp-- is equivalent to that of the Re radicals randomly distributed around the Bp-. with the concentration of 0.53/0.307 = 1.7 mmol/dm3. Assumption of the Bp-s-R. Pair. Let us take an alternative assumption that the longitudinal relaxation of each Bp-. radical anion in the y-irradiated samples is caused solely by its counterpart R. radical formed as a pair with itself. From a radiation-chemical point of view, this is equivalent to assuming that all the spurs surviving at the time of measurements are isolated from each other and contain only one ion pair. The recovery kinetics is given by the superposition of the relaxation behavior of Bp--R. pairs having a different intrapair distance, r, and a different angle, 8, as V(t) = Vo[ 1 - x w d r x T 2 ~ r 2 d (exp(-Af6t( r) 1 - 3 cos2 8)2) sin 8 do] (15) where d(r) is the distribution function of the distance between Bp-. and R- in a pair. Since the exponent of t for the recovery kinetics is the same for all the R- concentrations examined, it is plausible to assume that the theoretical recovery kinetics by the excitation transfer to the paired R- are the same as those to the Re radicals randomly distributed around the Bp--, that is, V(t) = Vo(l - e~p(-bt’/~)}.The deviation of the exponent of t from the theoretical value of to the experimental value of 0.82 may also arise from the flip-flops between the excited spins. The distribution function giving the exponent of t close to is given by @(r) = 3 [ e ~ p ( - r ~ / r , 6 ) / ( 2 ~ ~ / ~ r ~ ~ )(16) ] which gives for the exponent of t if the angular term in eq 15 is ignored. With the above distribution function, eq 15 can be numerically integrated to give the recovery kinetics of the echo intensity shown in Figure 6 (solid curve). It does not follow an exponential behavior as expressed by a straight line in the figure. However, it can be approximated by the same functional form as eq 13, i.e. V(t) = Vo(l - e ~ p [ - f ~ r , - ~ ( A t ) ~ / * ] ]
rir, Figure 7. Distribution function for the intrapair distance of the Bp-s-R.
(17)
pairs. The arrow indicates the average intrapair distance. Parameterfis determined from Figure 6 to be 0.94-1.14 in the region of V(t)/Vo = 0.1-0.9. By comparing eq 13 with eq 17, one can correlate ro with the “residual” concentration, as ro = ( 9 / ( 1 6 ~ ) ) 1 / 3 ( 3 / ~ ) 1 6 f cThe ’ / 3 .average distance between the Bp-. radical anion and the Re radical generated in pairs by y-radiolysis is obtained from r, to be 4.4 k 0.4 nm from the residual concentration of 1.7 mmol/dm3. The spatial distribution of the paired radicals is shown in Figure 7. Intrapair Distance of the el,--R. Pair. The intrapair distance of the el,--R- pair could not be estimated in the present study, because the “intrinsic” relaxation of elo; free from other paramagnetic entities is unknown. However, it seems to be similar to that for the Bp-sR. pair, because the longitudinal relaxation for elo; is almost the same as that for Bp-.. Smith and Pieroni examined early in 1965 the ESR power-saturation behavior of e,, in the glassy M T H F matrix and estimated the local concentration of paramagnetic entities in spurs to be 20 mmol/dm3,* which is more than 10 times higher than the present estimate, 1.7 mmol/dm3. This discrepancy comes essentially from the assumption in the previous study that the transverse relaxation is due only to the electron spin-spin interaction. The present study has shown that the electron spin-nuclear spin interaction contributes equally to the transverse relaxation of e],- (see Figure 2 and eq 2 ) . Lin and Kevan3 also studied the paramagnetic relaxation of eloc-in the M T H F matrix by the power-saturation method and reported that the product of the relaxation times, TIT2,was independent of the radiation dose up to 1.4 Mrd. They estimated the upper limit of the spur radius to be 6.3 nm by assuming that the spurs began to overlap with each other at this dose. This estimate seems consistent with the present result on the average intrapair distance, 4.4 nm. The present investigation has shown that the rates of both the longitudinal and transverse relaxations are dependent on the radiation dose in the whole range down to zero, though the relaxation times, T , and T2,cannot be uniquely defined for elo; (also for Bp-e) in the rigid matrix because of various magnetic environments. Such a dose dependence may have been lost in the technical difficulties of measurements and the complexities of analyses in the previous power-saturation study. Acknowledgment. This work was in part supported by the Ministry of Education and Culture under Grant No. 56430026.
