J. Phys. Chem. 1980, 84, 1253-1258
1253
Photoioni2!ation in Nonpolar Liquids Studled by Electric Field Quenching of Recombination Fluorescence and Photoconductivity J. Bullot,' P. Cordier, Laboratoire de Physico-Chimie des Rayonnements, UniversU Paris-Sud, Centre dOrsay, 9 1405 Orsay Cedex, France
and M. Gauthier ERA 718, Universif6 Paris-Sud, BAt 350, 91405 Orsay, France (Received July 17, 1979) Publication costs assisted by CNRS
It is shown that the combined measurements of the electric field quenching of fluorescence, AFIF',and of the photocurrent, i, give a complete set of information on geminate pair creation, electron escape, and electrorrion recombination in photoionized nonpolar solutions. First a model is proposed which predicts a linear relationship between AF/F' and i, in very good agreement with experimental data obtained in solutions of TMPD in tetramethylsilane (Me4Si),2,2-dimethylbutane (DMB), 2,2,4-trimethylpentane (TMP),and cyclopentane (cP). 1.71 X 2.25 X From the AF/F' vs. i plots, the zero-field quantum yield is obtained: 02 = 7.58 X in Me,Si, DMB, TMP, and cP, respectively. Also the zero-field current i is measured and 0.42 X from such plots. Then use is made of the Oinsager theory for calculating the field dependence of the electron escape probability P(E) in these liquids. For this purpose one-parameter Gaussian and truncated exponential distribution functions of initialdistances are introduced and (ilio)experimental data are numerically fitted to theoretical P(E)/P(E=O)ratios. Characteristic distance parameters are obtained e.g., for a Gaussian function: 172,106, and 116 A in Me4Si,DMB, and TMP, respectively. The knowledge of &" and P(E=O) yields the so far unknown quantity $qp, the quantum yield for geminate pair creation: $ip = 0.30,0.19, and 0.12 in Me,Si, DMB, and TMP in the case of a Gaussian distribution function. Finally we give an early account of how the method can be used for studying the photon energy dependence of &", &, and of the distance parameters. A salient feature is that both :4 and (hip go through a maximum at a photon energy -5.7 eV.
1. Introduction
To date most of the experimental information on charge carriers creation and recombination in nonpolar liquids comes from either conductivity studies or electron scavenging in photoionized dilute solution^^-^ and in pure liquids exciteid with high-energy radiation.&' Experiments on field and temperature dependence have been carried out and a picture of electron escape mainly based on the Onsager theoryE has emerged. All these investigations share in comimon the fact that they are dealing with free electrons only, i.e., with electrons escaping the strong Coulomb field of the parent ions. As in these low permittivity liquids the local field around an elementary charge is extremely high the application of an external electric field induces a very small perturbation indeed. In this type of study the highest electric field strengths attainedg are of the order of 200 kV cm-l which corresponds to the value of the Coulomb field at a distance r = 60 A in a liquid of dielectric constant e = 2. It should be noted that the field strength cannot be much increased, for example in n-hexane breakdown has been observed at -400 kV cm-l.lo Then only the electrons which are ejected at large distances from the parent cation have a nonnegligible probability of escaping recombination. It is known that the electron range greatly depends upon the nature of the liquid5p6but even in the most favorable cases, i.e., in liquids in which the electron is quasi-free, electron-ion recombination remains the most probable event and only a very few geminate pairs separate to give free carriers. In a way it may be said that data obtained from free carriers only represent partial information on the ionization process. In particular no information has been obtained on the primary process: the geminate pair creation, its quantum yield n,& pair (&), and the real shape of the distribution function a(r,X)of the initial separation distance r.'P This
is evidenced by the fact that, when using the Onsager cheory of electron escape probability in the presence of an externalfield, conductivity datgalone do not allow one to clearly distinguish between various a(r,X), e.g., Gaussian or exponential. This is due to the fact that most of the electrons picked up by the field are located at large r in the tailing part of a(r,X) whereas the interesting feature of a(r,X)should be found at relatively short r. A clearcut distinction would need the use of much higher field strengths than those attained in the laboratory. A way of recovering the lost information is to look at the geminate recombination itself and this has been achieved by studying the electric field dependence of the recombination fluorescence emitted by a dilute solute of a photoionizable m~lecule.l'-~~ When photoionization is a competitive process with "normal" fluorescence obtained by internal conversion to the first singlet state, i.e., if c#;~ is large enough, we have shown that if an external electric field of sufficiently high strength E is applied a quenching of the fluorescence is observed.ll The fluorescence intensity P normally seen at E = 0 is decreased b y an amount A F which depends upon the field strength. This effect has been mainly studied with TMPD (N,N,N',N'-tetramethyl-p-phenylenediamine)in some nonpolar solvents but is expected not to be limited to this solute. Indeed a similar though much weaker effect, has also been detected in indole-tetramethylsilane (Me4Si) solutions. This way of recovering the information may be extremely valuable; AF/P values as large as 0.18 have lbeen measured at room temperature with TMPD-Me4Si iiolutions excited at 217 nm (5.71 eV) and submitted to an electric field of 70 kV cm-l. Thus direct informations on the recombination process are obtained, and when combined with the classical photoconductivity measurements 0 1980 American Chemical Society
1254
The Journal of Physical Chemistry, Vol. 84, No. IO, 1980 s3 s2
Bullot, Cordier, and Gauthier
On this basis the zero-field fluorescence intensity can be written as
P
= K'[(1
Po) Chipd'F
(1- $'ip)$'F]
(5)
Whereas in the presence of an electric field one has
F = K l ( 1 P)$ip$F + ( 1 - $ip)@F]
(6)
where K'is an experimental factor. Therefore the relative fluorescence intensity decrease (F'- F)/Pis
Figure 1. Model of electron escape and recombination depicting the origin of the photocurrent and the dual character of fluorescence emission.
we have in hand a complete set of information. In what follows we present a thorough account of the method. First we discuss the model and the equations linking the photocurrent and the relative decrease of the fluorescence intensity AF/P to the physical quantities related to photoionization. Then these equations are used (1) for obtaining from straight experimental data the zero-field free electron quantum yield 4: and the zero-field photocurrent io, (2) for computing by means of the Onsager theory the distance parameter h of a given distribution function a(r,h) and the quantum yield for geminate pair creation $ip. Finally, the described method is used for giving an early account of the solvent and photon energy dependence of these quantities. All experimental details can be found in previous reports.11J2
2. Relationship between Photocurrent and Fluorescence Quenching Measurements In Figure 1 we present a Jablonski-type diagram for the photoionization of the solute molecule TMPD in a nonpolar liquid. Absorption by TMPD of one photon whose energy is larger than the photoionization threshold in a given ~ o l v e n tleads ~ ? ~ to the formation of a geminate pair [M+,e-]with a quantum yield hPand to internal conversion to the fiist single state followed by emission of fluorescence with an efficiency (1 - +ip)$F (1) where $F is the fluorescence quantum yield. Here it is assumed that the TMPD concentration is sufficiently low for occurrence of separate noninteracting ion pairs. When an external electric field of strength E is applied the electron escape probability is P and a photocurrent i is measured at the electrodes (2) i = K& = Kqji$ where is the free electron quantum field at E and K is an experimental factor. However, most of the geminate pairs undergo recombination to give excited TMPD molecules which subsequently emit the so-called recombination fluorescence with an efficiency (1-- P)@ip& (3) At this point it should be emphasized that at E = 0 the thermal escape of electrons from the parent ion Coulomb field is still possible with a nonzero probability Po and free electrons are produced with a quantum yield $2. It corresponds a theoretical zero-field current of (4) io = K$,O = K4iJ'O Of course io cannot be measured because at very low fields (below some hundreds of V cm-') the charges no longer drift toward the electrodes.
From eq 2, 4 and 7 one deduces the most important relationship
linking the two sets of measurements. It may be noted that the fluorescence quantum yield +F introduced in expressions 1and 3 disappears in the final equations 7 and 8 describing the field effect. This is so because the field effect is due to photoionization which occurs at energies far above the first singlet state S1 in nonpolar solutions. In contrast, 4Fwhich accounts for the nonradiative processes occurring from S1is not expected to be field dependent. It has been experimentally checked that the fluorescence intensity does not depend on the applied electric field when the excitation wavelength is larger than the photoionization threshold." The proposed model is based on several assumptions. (1)Photoionization is a one-photon process. This has been demonstrated by Jarnagin and co-workers1 and confirmed below. (2) Only two deactivation paths for the primary photoexcited state are considered: namely, photoionization and internal conversion toward S1. Other possible mechanisms, such as decomposition, intersystem crossing, or direct internal conversion toward the ground state, are assumed to occur only from S1and are responsible of the low & value (& 0.14 in Me4Si and 0.19 in TMP or nHex14). If any of these processes originates from a state above S1, with a given energy threshold Eth, then a significant decrease of 4 F would be observed at energies E > Eth. This is not observed experimentally; it has been shown that q5F depends only slightly on the excitation energy at least up to -6.2 eV (200 nm).4114915 (3) It is assumed that the spin of the photoelectron remains correlated with that of the cation during the lifetime of the geminate pair, so that geminate recombination only yields singlet states. Such an assumption is supported by the theoretical analysis of Brocklehurst.lG Furthermore, if the ion pair lifetime is long enough for spin relaxation to develop, then recombination would yield mainly nonfluorescent triplet states, and this would again lead to a decrease of 4Fat energies above the ionization threshold, provided 4ipis not negligible. Indeed the high AF/P values observed and the calculations developed below show that q5ip may be quite important for excitation wavelengths in the 2W220-nm range. Then the fact noted above (that 4Fis not affected) yields experimental evidence that spin correlation is preserved in the geminate pair. (4) It is assumed that the quantum yield for geminate pair creation 4ipdoes not depend upon the field strength E. If so a shift of the photoionization threshold Ifiqwould be expected. We have investigated this point by measuring the excitation wavelength dependence of the photoconductivity in a -2 X M TMPD-Me4Si solution at E
-
Photoionization in Nonpolar Liquids
The Journal of Physical Chemistry, Vol. 84, No. 10, 1980 1255
0.15 AF
7 AF
0.1(
0.0: 0.05
1
3
2 lXIO-lo
4
(A)
Experimental check of eq 8: (0)= Me.& (A)= DMEI; (A) = TMP; (*) = cP. Same concentration, irradiation,and temperature conditions as in Figure 3.
Figure 2.
Figure 3.
= 4.69 kV cn1-l and at E = 40.6 kV cm-', following the previously described m e t h ~ d In . ~ bath cases the same Iliq = 4.47 f 0.04 eV was measured. We then conclude that at least in the medium field strength range $ip is constant. It is interesting to note that if at very high field strengths @ip becomes field dependent, though 2, 4, and 7 are no longer valid, eq 8 remains valid. This is easily shown by taking &" at E = 0 and &,(E) at E # 0. Equations 2,4, and 7 now read i = K& = K4i,(E)P (2')
TABLE I: Experimental Data at hue,, = 5.90 eV and T = 293 K a F / F Qvs. i plots interfl ,e* AF/Fo cept (cm ( E = 70 slope X @eOX V-I liquid kV cm-I ) x 10' Fz s-,' 7.58 90 9.02 8.20 0.999 Me$ 0.153 2.25 10.9 8.91 2.30 0.989 DMB 0.067 1.71 7 4.63 1.74 0.993 TMP 0.059'" 1.1 16.7 0.42 0.911 -0.42 cPb 0.012 Extrapolated value, breakdown in the gas phase above the solution occuring at E 30 kV cm-'. As shown in Figure 4 CPdata are very inaccurate because AFIFO at low field strengths is of the order of our present experimental accuracy on A F / F o : f 2 X From ref 6.
Electric field quenching of fluorescencein various liquids. For all solutionsTMPD N 3 X M, hvexc= 5.90 eV, and T = 293 K: (9) and (0)= Me,Si (two runs); (A)= DMB; (A) = TMP, measurements end at E = 28 kV cm-' where breakdown occurs in the gas phase above the solution; (*) = cP, for the sake of clarity average values are shown, data scattering being indicated in the insert of Figure 4.
io = K4,O = K&i,OP
(4') (7')
Elimination of P between these equations yields
which does not depend upon 4i,(E) and is identical with (8). 3. Results arnd Discussion 3.1, Measurement of Zero-Field Quantities from Straight Experimental Data. The zero-field quantities 4: and io are easily obtained from eq 8 if as expected a linear relationship between AF/P and i is experimentally observed. 4:' is deduced from the intercept &"/(l-42) whereas io is given by the abcissa of the point of zero ordinate. So far we studied four solvents: tetramethylsilane (Me&), 2,2.,dimethylbutane (DMB), 2,2,4-trimethylpentane (TMP), and cyclopentane (cP). In all of them a linear relationship between AF/P and i is found under various photon energy excitation flux and temperature conditions even in liquids for which both the AFIP vs. E and the i vs. E characteristics are curved. In Figure 2 is shown the field dependence of AF/P and in Figure 3 the AF/P vs. i plots for the four liquids irradiated at room temperature vvith photons of 5.90-eV energy. As seen good straight lines are obtained. In Table I we present the slope, intercept, and correlation coefficients P2 of the regression lines, together with eF/P values at E = 70 kV cm-l and
~
-
the zero-field quantum yields $2measured from the intercept values. We do not present the i-E characteristics because the experimental proportionality factor K in eq 2 varies from one experiment to the next; instead using the io values measured from the plots of Figure 3, we Fihow below the ratio i/iowhich permits a direct comparison of the behavior of these liquids (Figure 5 ) . The possibility of measuring io was used for testing the monophotonic character of TMPD photoionization. For this purpose we studied the dependence of AF/P and i on the excitation photon flux 3. Obviously, at a given excitation energy and temperature when 3 is varied $2 should remain constant whereas iowhich is proportional to the free charge density should vary linearly as
io = Ar#+,3P (9) where A is an experimental factor. This point was checked in a 2 X M TMPD-Me4Si solution irradiated with 5.90-eV photons at 294 K. The light intensity was varied by a factor 10, with the maximum flux measured at the exit slit of the deuterium lamp-monochromator assernbly being 3.7 X 10" photons s?. In Figure 4 are shown the AF/P vs. i plots. Good straight lines are obtained and as anticipated they all converge to very close intercepts yielding 7.3 X I4: I8.02 X In the insert of Figure 4 is shown the photon flux dependence of io. A linear dependence is found confirming the one-photon nature of the photoionization. Returning to eq 7 which we may write as u / p = ( 4 e - 42)/(1 - 42) -4
1258
The Journal of Physical Chemistry, Vol. 84, No. 10, 1980
Flgure 4. A F / P vs. ivariations as a function of the excitation photon flux 9 in Me,Si-TMPD at hv,,, = 5.90 eV, T = 294 K. The number along each line is proportional to 3.
it is seen that knowing $2and the field dependence of AF/P one may obtain the field dependence of the free electron quantum yield $,(E) as
$,(E) =
Flgure 5. Numerical calculations. Comparison of experimental i l i o data to P(E)/P(E=O) ratios calculated on the basis of the Onsager theory: (0)= Me4Siexpt., (-) = Me4Sitheory; (A) = DMB expt, (---) = DMB theory; (A) = TMP expt, (-.-.) = TMP theory; for the sake c h r i w d a t a are shifted upward by 0.5 unit. The calculated P(E)IP(E=O) ratlos refer to the Gaussian distribution function (eq 14).
TABLE 11: Numerical Calculations at hu,,, = 5.90 eVandT=293K GAUSS
(m/J9I + m/P
(10) An example of such results has been previously presented.12 It must be emphasized that eq 8 does not depend upon any theory of electron escape from the counterion Coulomb field in the presence of an electric field. Two consequences are deduced: (1)Equation 8 may serve to test the validity of any theory giving the field dependence of the probability ratio: PIP" = g(E). As PIP" = i/io it is seen from eq 8 that -
Bullot, Cordier, and Gauthier
Then when plotting AFIP against g(E), the theory under test must be such that (i) the plot is linear and (ii) the ratio R = (-slope/intercept) = 1. This is what we call the "Rtest". The Poole-Frenkel model has been R-tested.17 (2) The measured zero-field quantities may be used with confidence. In particular the exact knowledge of io is quite important to carry out the numerical calculations we are now to discuss. 3.2. Calculations Based on the Onsager Theory. This theory8 has proved very successful in explaining the charge pair separation and recombination in a variety of systems: amorphous selenium,18 anthracene single crystal^,^^-^^ doped polymeric mat rice^,^^^^^ and, in particular, in pure or doped nonpolar l i q ~ i d s . ~ The ~ ~original -~ formulation of Onsager has been much worked out so as to render it amenable to numerical calculations. In particular Terlecki and F i ~ t a ktaking , ~ ~ into account the fact that electrons are isotropically ejected around the cations, derived the space-averaged escape probability P(r,E). As P(r,E) depends upon the initial distance r separating the ions, a distribution function a(r,X) has to be introduced, h being a distance parameter, and the average escape probability reads
P(E) = 4 ~ ~0~ r ' a ( r , h ) P ( rdr ,E) E
# 0
(12)
In order to determine the value of X the usual procedure consists in fitting the calculated ratio P(E)/P(E=O)to the
liquid Me,Si DMB TMP a
r, 172 106 116
PT) 0.25
Exp
@b
b
P(0)
0.30 158 0.14 0.12 0.19 66 0.04 0.14 0.12 72 0.05
@b
rma
0.54
158.9
0.56 0.34
92 95
From ref 27.
experimental current ratio i/iO. Extensive calculations along these lines have been carried out in the past few years mostly in pure liquids irradiated with high-energy particle^.^?^ As said before various distribution functions were tested: Gaussian, truncated exponential, Dirac, power-tailed. However, whatever a(r,h) a good fit is always obtained and on this sole basis it is extremely difficult to sort out the right distribution function even when high field strengths -200 kV cm-l are used.g Anyway this calculation step is necessary to get, for a given a(r,X), the distance parameter X and the related field dependence of the electron escape probability P(E). We carried out such calculations on the above experimental data (see also ref 13). We chose two realistic &,A): the one-parameter Gaussian distribution function (GAUSS):
and the one-parameter truncated exponential distribution function (EXP): ifrla o(r,h) = 0 if r < a (15) In the latter case the parameter a was kept constant and equal to 5 A as suggested by Abell and Funabashi.26 The results of these calculations are shown in Figure 5 for Me,Si, DMB, and TMP in the case of a GAUSS function. As anticipated the EXP function also gives a very good fit to the experimental i/iOdata. It should be noted that as CPmeasurements are very inaccurate, no calculation was performed for this solvent. In Table I1 are shown the corresponding distance parameters r, and b and the calculated zero-field electron escape probabilities P(E=O). Now as $2has been previ-
The Journal of Physical Chemlstv, Vol. 84, No. 10, 1980 1257
Photoionization in Nonpolar Liquids
ously determined it is seen that the quantum yield for geminate pair creation 4ipmay be obtained: $ip = 4:/ P(E=O). Of course as P(E=O) depends upon the type of distribution function chosen different 4ipvalues are thus obtained (see Table 11, columns 4 and 7). It should be emphasized that if q5ip could be otherwise determined it would be poissible to decide according to what type of distribution function the ejected electrons are thermalized around the cations. We have shiown previo~sly'~ that a good fit of the M/F' - E characteristics is obtained by means of eq 7 and by adjusting the parameter dip. The same values are found. 3.3. Solve,nt and Photon Energy Dependence of the Photoionization Physical Parameters. The method just described has been used to study how the quantum yields $2,d,(E), 4ip,and the distance parameters r, and b vary with the solvent or the excitation photon energy hv,,,. 3.3.1 Solvent Effect. The results concerning the effect of solvent have been given in Tables I and 11. As expected and in agreement with previous studies2the higher the electron moblility in a given solvent, the larger $2 in the same liquid (compare columns 6 and 7 of Table I). The calculated values in Table I1 show that whatever the type of distribution function $2 variation results from the simultaneous decrease of dipand P(E=O). It should also be noted that the calculated distance parameters are of the same order of magnitude as those previously reported when pure solvents are excited with high-energyr a d i a t i ~ n As .~~~ far as the Gaussian distribution is concerned r, is very close to the values found by Schmidt and Allen2' (last column of Table 11). 3.3.2. Excitation Photon Energy Dependence. The effect of the excitation photon energy was carefully examined. Many experiments were carried out with Me4Si and DMB solutions and only the salient features are presented below. The available energy range is limited. At low energy AF/P decreases drastically and drops below our experimental accuracy (- f 2 X lom3)at a photon energy -0.5 eV above the photoionization energy threshold (4.45 eV in Me4Si anti 4.80 eV in IlMB).3 The highest photon energy of practical use is limited by solvent absorption. For the -3 :K M TMPD-Me4% solutions used, the solvent absorption becomes comparable to the TMPD absorption at -6.2 eV. On the other hand, the solute concentration cannot be increased because volume recombination effects would render our measurements meaningless. The A F / P and i action spectra were run at E = 39 and 4.69 kV cm-l, respectively. It has been found that changing the field strength does not alter the qualitative features of the spectra. In both Me4Si and DMB solutions we observed that AF/P goes through a maximum at hue,, N 5.7 eV (217 nm) and increases for hv,, k 5.9 eV (210 nm). The action spectra of the photocurrent show the same features. The photon energy dependence of $2 is presented in Figure 6 in the case of a TMPD-Me4Si solution. Each of the points plotted was obtained through a complete field strength study of AF/F' and i according t o the method described above. Here again a maximum at 5.7 eV clearly appears, together with an increase at high energy, The numerical calculations show that rm and b (the Gaussian and exponential distribution functions parameters) vary only slightly with the excitation energy. The are roughly constant for hv,,, k 5.4 eV (r, 160-180 and b 140-160 A) but significantly lower values have been obtained at 5.1'7 eV (rm= 138 A and b = 104 A). In
-
-
K
X exc(nm) 240
r~
O*'t
0
230
220
210
200
I
I
I
I
// 1
1
0
1258
J. Phys. Chem. 1980, 84, 1258-1259
Acknowledgment. Professor S. Lipsky and Mr K. Lee are gratefully acknowledged for having made available to us recent results and for helpful comments.
However, from the results obtained in n-hexane these authors suggest that another deactivation path (quantum fleld dn)might occur from the third singlet state of TMPD (below about 218 nm). To take it into account eq 7 and 8 must be rewritten as
References and Notes (1) R. C. Jarnagin, Acc. Chem, Res., 4,420 (1971);S.H. Peterson, M. Yaffe, J. A. Schultz, and R. C. Jarnagin, J . Chem. Phys 63, .1
2625 (1975). (2) R. A. Holroyd and R. L. Russell, J . Phys. Chem., 78,2128 (1974). (3) J. Bullot and M. Gauthier, Can. J . Chem., 55, 1821 (1977). (4) K. Wu and S.Lipsky, J. Chem. Phys., 66,5614 (1977). (5) A. Hummel, Adv. Radiat. Chem., 4, 1 (1974). (6) H. T. Davis and R. G. Brown, Adv. Chem. Phys., 31,329 (1975). (7)J. P. Dodektet,Can. J. Chem.,55,2050 (1977),and references therein; J. P. Dodelet, K. Shinsaka, and G. R. Freeman, Can. J . Chem., 54, 744 (1976),and references therein. (8) L. Onsager, Phys. Rev., 54, 554 (1938). (9) J. Casanovas, R. Grob, D. Bhnc, G. Brunet, and J. Mathieu, J . Chem. Phys., 63, 3673 (1975);J. Casanovas, Thesis, Toulouse, 1975. (10) P. Wong and E. 0.Forster, Can. J. Chem., 55, 1890 (1977). (1 1) J. Bullot, P. Cordw, and M. Gauthii, Chem. phys. Left., 54,77 (1978). (12)J. Bullot, P. Cadi, and M. Gauthii, J. Chem. phys., 69,1374 (4978). (13) J. Buliot, P. C a d i , and M. Gauthii, J. Chem. phys., 69,4908(1978). Equatlon 5 of this paper shoukl read as u(r,E,O) instead of u(r,E,B). (14) M. Gauthier, R. Klein, and I.Tatischeff, unpublished results. (15) The slight + F decrease observed In Me,Si and 2,2,4-trimethylpentane in the absence of an external electric field has been accounted for quantitatively by the production of free electrons by Wu and L i p ~ k y . ~
From ref 4 9 Dcannot be hlgher than 0.05 for A 5210 nm (E,,, 2 5.9 eV). In this wavelength range, our 4: determinations, and hence 9 ip, would thus be underestlmated by -5 % . (16) 6.Brocklehurst, Nature(London), 221, 921 (1969);Chem. Phys. Lett., 28, 357 (1974). (17) Yu A. Berlin, J. Bullot, P. Cordier, and M. Gauthier, Radiat. Phys. Chem., in press. (18) D. M. Pai and R. C. Enck, Phys. Rev. B , 11, 5163 (1975). (19) R. H. Batt, C. L. Braun, and J. F. Hornig. Appl. Opt. Suppi., 3, 20
(1969). (20) R. R. Chance and C. L. Braun, J . Chem. Phys., 64,3573 (1976). (21) L. E. Lyons and K. A. Milne, J . Chem. Phys., 65, 1474 (1976). (22) P. M. Borsenberger and A. I.Ateya, J. Appl. Phys., 48,4035(1978). (23) P. M. Borsenberger, L. E. Contois, and D.C. Hoesterey, J . Chem. Phys., 68,637 (1978). (24)J. Terlecki and J. Fiutak, Int. J. Radiat. Phys. Chem., 4,469 (1972). (25) J. Ortet, These 3e cycle, Toulouse, 1978. (26) G. C. Abell and K. Funabashi, J . Chem. Phys., 58, 1079 (1973). (27) W. F. Schmidt and A. 0. Alien, J . Chem. Phys., 52,2345 (1970). (28)S. Llpsky, private communication.
Symmetry Properties of the Transport Coefficients of Charged Particles in Disordered Materials James K. B a l d Health and Safety Research Division, Oak RMge National Laboratory, Oak Ridge, Tennessee 37830 (Received July 17, 1979) Publication costs assisted by the Oak Ridge National Laboratory
The transport coefficients of a charged particle in an isotropic material are shown to be even functions of the applied electric field. We discuss the limitation which this result and its consequences place upon formulae used to represent these coefficients.
The mobility of charged particles in isotropic materials (gases, liquids, and amorphous solids) is a scalar. If the drift velocity is a nonlinear function of the electric field E , then the mobility p is a function of E also. Because of either charge conjugation or parity invariance,1*2 p must be an even function of E , namely, p ( E ) = p(-E). Thus, if f(E) is a function containing empirical parameters to be fitted to mobility data, we must have p ( E ) = (1/2)[f(E) 4- f(-E)I (1) The function f(E)must be real valued for negative values of its argument. Below, we discuss some examples. The semiempirical formula p ( E ) = p ( 0 ) ( e X E / k ! V 1sinh (eXE/kn (2) has been used to represent the mobility of electrons which proceed through a material by the process of thermally activated h ~ p p i n g .Here, ~ p(0) is the zero field mobility, X the jump distance, e the magnitude of the electron charge, h Boltzmann’s constant, and T is the absolute temperature. This formula can be constructed from eq 1 by letting f(E) = p(O)(eXE/kT)-l exp(eXE/kT). In the case of the drift of He+, Ne+, and Ar’ ions in their parent gases, the mobility goes over smoothly from a value ‘Also adjunct member of the faculty of the Department of Physics, University of Kansas, Lawrence, KS 66045. 0022-3654/80/20841258$0 1 .OO/O
p(0) at zero field to p ( E )
-
at high field. To represent this behavior, Frost suggested the empirical formula P(E) =
d O ) P + a(E/P)l-”z
(3)
where p is the neutral gas pressure and a is an adjustable parameter! This formula is not an even function of E and, as such, is not acceptable. A formula with the proper symmetry which gives the same limiting behavior is P(E) = p(0)[1 + a(E/p)Zl-’/4
(4)
It is not in contradiction with the invariance principles for the mobility to be approximately proportional to E-’/2at high field. Rather, it means that the term c ~ ( E / pis) so ~ much larger than unity in the factor 11+ ~ ( E / p ) ~ ] that -l/~ the experiment is not capable of sensing the complete formula. It is not acceptable to attempt to salvage eq 3 by replacing E by IEl since the resulting formula is not analytic a t E = 0.l Finally, it is to be noted that, when the mobility is field dependent, the longitudinal diffusion coefficient D&E)is Again, because of the invariance principles, we must have D,,(E) = Dll(-E). Acknowledgment. This research was sponsored by the Office of Health and Environmental Research, U.S. Department of Energy, under contract W-7405-eng-26 with 0 1980 American
Chemical Society
