One‐photon intrinsic photogeneration in anthracene crystalsL. E. Lyons and K. A. Milne Citation: The Journal of Chemical Physics 65, 1474 (1976); doi: 10.1063/1.433201 View online: http://dx.doi.org/10.1063/1.433201 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/65/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Mechanisms in environmentally assisted one-photon phase control J. Chem. Phys. 139, 164123 (2013); 10.1063/1.4825358 Solvent effects on the vibronic one-photon absorption profiles of dioxaborine heterocycles J. Chem. Phys. 123, 194311 (2005); 10.1063/1.2121590 Photogeneration of free carriers near 4000 Å at 77 °K in anthracene crystals J. Chem. Phys. 62, 3269 (1975); 10.1063/1.430879 Intrinsic Photoconduction in Anthracene Crystals J. Chem. Phys. 45, 3966 (1966); 10.1063/1.1727446 APL Photonics

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One-photon intrinsic photogeneration in anthracene crystals L. E. Lyons and K. A. Milne

Chemistry Departmellt. University of Queensland. St. Lucia 4067. Queensland. Australia (Received 12 January 1976)

Steady state one-photon intrinsic photogeneration in anthracene crystals has been measured using I M Na,SO.l aqueous solutions as electrodes, as a function of photon energy and electric field (up to 6 X \07

V m --'). Anomalies in previous measurements with aqueous electrodes have been resolved. The values of absolute photoconduction yield now agree for aqueous, blocking, and metal electrodes, and for pulsed and steady state measurements. The Onsager theory of geminate recombination explains the shape of the current-field dependence better than does a theory of Knights and Davis. Interpretation of the current-field curves in terms of Onsager's theory yields values of ro (the initial separation of the charges in the ion pair) as a function of photon energy and dielectric constant, assuming a delta function distribution for roo Two ion pair states appear to be formed: at hv = 3.9 to 4.2 eV, rOA = 1.8 nm (for E, = 3.8); at 4.5 to 5.0 eV, rOB

= 2.5 nm. For hv=4.2 to 4.5 eV, ro undergoes a smooth transition. The maximum value (0.1) of </>" the one­photon ion pair production efficiency, occurs at 4.2±0.15 eV, near the maximum in Tj, the two-photon ion pair yield. The agreement of peaks in </>, and Tj with a peak at 4.1 eV in O"~-, the two-photon absorption cross section, is discussed. From the energy dependence of </>, and the magnetic field effect on prompt fluorescence, it is concluded that, in the process of singlet exciton fission into two triplet excitons, the lowest ion pair state is not the fissionable singlet. Only a very small [O.07(±0.03)%], direction independent, magnetic field effect on the intrinsic photocurrent was observed (at a photon energy of 4.5 eV).

I. INTRODUCTION

The intrinsic photogeneration of electrons and holes in single crystals of anthracene by a one-photon process was discovered by Castro and Hornig in 1965. 1 Subse­quent investigations, however,2 have failed to clarify the primary process in the formation of free charge carriers. There has been no direct evidence for the presence of a charge transfer state in the crystal, and no direct optical transition to the conduction band has been observed in the one-photon absorption spectrum, The term "autoionization,,3 has been used to refer to the process by which light energy, absorbed in the vi­bronic manifold of an excited singlet state, produces a pair of charge carriers.

The dependence of the one-photon intrinsic genera­tion rate on electric field strength has been studied by Batt, Braun, and Hornig, 4,5 Geacintov and Pope, 6 and Chance and Braun. 7-9

Batt, Braun, and Hornig, 4.5 using blocking contacts, found that up to 2x 106 V m-1, the intrinsic photocurrent varied linearly with field, and the slope/intercept ratio of the current-field plot was in near quantitative agree­ment with that predicted by the Onsager theory of gemi­nate recombination. 1o They also measured the activa­tion energy of the intrinsic photocurrent as a function of photon energy and, hence, assuming a Coulombic po­tential well, calculated values of Yo, the separation of the electron and hole in the initially generated elec­tron-hole pair. They reported that Yo= 8. 5 nm at pho­ton energies of 4.2 to 5.1 eV and Y o=13.8 nm at 5.8 to 5.9 eV r Er (the relative dielectric constant) = 3.2]_

Geacintov and Pope, 6 using water, aluminium, and blocking contacts, found good agreement in absolute photoconduction efficiency, between blocking and alumi­nium contacts, but for water contacts, the efficiency was an order of magnitude larger. Also, they extended the current-field relationships to higher fields than did Batt, Braun, and Hornig, and found poor agreement

with the Poole- Frenkel and Schottky effects and also with the Onsager geminate recombination theory. They assumedYo=8.3 nm.

Chance and Braun,7 using blocking electrodes, con­firmed that for fields up to 2_ 5 x 106 V m-1, the intrinsic photocurrents are explained by the Onsager model, with quantitative agreement between the experimental and theoretical values of the slope/intercept ratio. They explained the deViation, at fields below 105 V m -1, of the photoconduction efficiency, from that predicted by the Onsager theory, in terms of diffusion controlled re­combination of free and trapped carriers. Their ab­solute quantum yield agreed well with the experiments of Geacintov and Pope using blocking contacts. Better measurements of the activation energy of photoconduc­tion, reported recently by Chance and Braun8

•9 yielded

Yo = 5 nm for photon energies of 4.4 to 5.2 eV, and Yo

=6.7 nmfor 5.4 to 6.2 eV (E r =3.2).

We have resolved the anomaly in the behavior of aqueous contacts and have studied the field dependence of the absolute quantum efficiency, 1>, up to 6 x 107

V m-1; information on the carrier generation process, in particular the initial separation distance Yo, was ob­tained. Magnetic fields were also used.

II. EXPERIMENTAL

Anthracene sublimation flakes [(001) plane; 10-50 J.lm thick] were grown under reduced pressure of ni­trogenll •12 using commerically available anthracene (99.999 mol %). The anthracene was supplied under argon and not exposed to air until after the crystals were grown, the sublimation tube being charged with anthracene in a nitrogen atmosphere in a glove box. Before our purification procedure, the nitrogen was 99.99% pure with the level of O2 being less than 10 ppm (manufacturer's speCifications). This nitrogen was passed through a Commonwealth Industrial Gases Trace Oxygen Eliminator before entering the glove box. The

1474 The Journal of Chemical Physics, Vol. 65, No.4, 15 August 1976 Copyright © 1976 American I nstitute of Physics

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L. E. Lyons and K. A. Milne: Photogeneration in anthracene crystals 1475

T -2 0 oJ

'" :;:: z Q l-I) ::I 0 Z 0 I) 0 I-0 :J: n-o

r5 0 oJ

2 3 4 5 6

o

FIG. 1. Variation of photoconduction yield with E1/2. (E,

electric field strength). -- H20 contacts (see Footnote 6), hv =4.4 eV. - - - Al contacts (see Footnote 6), hv =4.4 eV. •......•. blocking contacts (see Footnote 7), hv = 4.9 e V. -'-'-' blocking contacts (see Footnote 6), hv =4.4 eV. -0 -0- present work: H20 contacts, hv = 4. 5 e V. -e-e-present work: Average of four crystals. Polyvinyl alcohol on back surface and aqueous l.OM Na2S03 contacts, hv =4.5 eV, (a) Poole-Frenkel slope (€r=3.8). (b) Schottky slope (€r=3.8).

Error bars indicate reproducibility of the yield-field relation­ship.

same purity nitrogen was passed through a column of BTS Katalysator at 323 K, to remove oxygen, an Ascarite column to absorb CO2, and a drying column of P20S' before being used to fill the sublimation tube. All operations with the crystals, except the thickness measurement, were performed in the dark or under yellow light. The extent of agreement of our results with those of other workers (see Figs. 1 and 2) indicates that the standard of crystal purity achieved by our meth­ods was sufficient to observe the intrinsic photogenera­tion unambiguously and in the absence of conflicting ex­trinsic processes (see Sec. Ill. A).

The crystal was mounted on one half of a sandwich type polystyrene cell with silicone grease, to give an effective area of exposed crystal of 0.07 cm2

• The thickness was measured1s with a calcite compensator in conjunction with a polarizing microscope. The a axis of the crystal was then determined14 by viewing the interference figure. A few drops of an aqueous emulsion of polyvinyl alcohoI1S

(a) were applied to the back surface of the crystal and allowed to dry. Then the cell was assembled, the electrode compartments filled with aqueous 1. OM Na2S0S solutions. 1S

(b) The whole cell was allowed to stand for several days. lS(e)

The electric field was applied, via gold wires dipping into the sulfite solutions, perpendicular to the ab plane of the crystal, by a constant voltage power supply, and an electrometer measured the voltage drop across a standard resistance in series with the electrically shielded cell. The crystal was illuminated through the Na~Os solutions on the side on which there was no polyvinyl alcohol, using a high intensity grating mono-

chromator and a deuterium light source. For po­larized light measurements we used a double Glan Taylor calcite prism polarizer.

Absolute photon flux measurements were made with a sensitive thermopile having a linear response down to 170 nm. In calculating the photon flux on the crys­tal, we allowed for reflection at all surfaces except the anthracene/Na2SOs solution interface, and took in-to account absorption by the sulfite solution around 260 nm. The effect of neglecting reflection is inSignificant in this region of the spectrum. 16 We calculated the yield of carriers of one sign pe r incident photon. Since all the light was absorbed in the crystal at the wavelengths used, this value was the same as the yield per absorbed photon.

The photon flux was varied using calibrated metal gauze screens.

A. Results The shape of the steady state yield-field relationship

at 4.5 eV studied on six crystals was superimposable above 10s V m -1 from crystal to crystal and also for one crystal. The error bars in Fig. 1 indicate the re­producibility of the relationship. Photo conduction yields for holes and electrons were equal at the highest fields, and at 106 V m -1 differed only by a factor of 2, the hole yield being larger. All results shown are for electrons.

Space charge did not limit the current at high or low fields. This statement is based on the following evi­dence: (0 Even at the lowest voltages, when the volt­age was changed during measurement of a current-volt­age relationship, no time decay of the photocurrent of more than about 10 s was observed. If the currents were space charge limited, then a time decay of the photocurrent of up to some minutes was expected, due to the rearrangement of trapped charge within the insula­tor. 17 (ii) A dependence of photocurrent on applied volt­age that was less than quadratic was not expected18 for

--/-..(nm)

FIG. 2. Variation of electron photoconduction yield with pho­ton energy. -- by E. Schmid in article by N. Karl, Adv. Solid State PhYSics 14, 261 (1974); 105-106 V m- 1; bare crystal; high vacuum. -0- 0- experimental results; polyvinyl alcohol film on back surface; l. 0 M NazSO:! contacts; thickness, 12 !-1m. field, 4x106 Vm-1•

J. Chern. Phys., Vol. 65, No.4, 15 August 1976

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1476 L. E. Lyons and K. A. Milne: Photogeneration in anthracene crystals

(C) (c)

(N)

FIG. 3. Side view of experimental arrangement for measure­ment of magnetic field effect on photoconduction. (a) light beam; (b) mirror; (c) magnet poles; (d) photoconductivity cell; (e) crystal mounted with ab plane horizontal; (f) cradle for photoconductivity cell; (g) leads to current- measuring equip­ment.

space charge limitation, irrespective of the energy dis­tribution of the traps. At fields less than 105 V m-t, a U I dependence was observed where 1 = 1. 0 to 1. 5 and U was the applied voltage. At fields greater fhan 105

V m-1 the current could not be described by a U l depen­dence. (iii) A low photon flux was used (2x 1012 pho­tons cm-2 S-1) and low photocurrents were observed ('" 10-10 A cm-2). Further points that support the ne­glect of space charge are (iv) the high fields used (i. e., greater than 105 V m -1) tended to remove space charge effects; (v) the yield-field relationship at low fields can be explained19 without invoking space charge limita­tion, by assuming bulk recombination of free and trapped carriers (see also Chance and Braun7). (At high fields, bulk recombination became insignificant and the photo­current measured the true photogene ration rate. )

Absolute measurements of photo conduction yields were made on four crystals and the average of these is shown in Fi.g. 1.

The current-field relationship was measured for pho­ton energies from 3.8 to 5.0 eV at 0.07 eV intervals. The measurements at 4.1, 4.5, and 4.9 eV agreed on two crystals (see Sec. m.B).

Figure 2 shows the absolute photo conduction yield as a function of energy at constant field. The observed photocurrent did not vary significantly with polarization of the light, and varied linearly with photon flux at high and low fields.

B. Magnetic field experiments Three crystals were used to investigate the magnetic

effect on the high electric field intrinsic photogene ration rate. TWO of these had been used for the current­electric fi.eld measurements. High fields were chosen to eliminate the effects of triplet detrapping of charge carriers, which enter into bulk recombination processes at low fields. 7.19 The experiment was set up as in Fig. 3, with the ab crystal plane horizontal. The cell was mounted in a Perspex frame. Light was reflected by an aluminum mirror mounted in brass, and focused by

a silica lens mounted on the Perspex frame. The cell could be rotated by hand about the vertical axis through its center. The field regulated electromagnet provided reversible magnetic fields from 0 to 10 kG.

When the cell was replaced with a nonmagnetic 108 n pressed powder resistor, the variation of the magnetic field produced only transient effects, of the order of 0.03%. The mains voltage, stabilized by a 100 W con­stant voltage transformer varied by less than 0.1% over the range of magnetic field.

A calibrated 1% variation in signal could be measured quantitatively, using a back-off circuit within the elec­trometer. The lower limit of reading was 0.03%.

Each experiment was performed by raiSing the mag­netic field from 0 to 10 kG over approx 1 min, leaving the current to stabilize at 10 kG for 2-3 min or longer if necessary, and then reducing the magnetic field from 10 to 0 kG over approx 1 min, again allowing the cur­rent to stabilize. The difference between the steady state zero reading measured before and after the field variation, and the 10 kG reading when stabilized, was taken as the steady state variation of current.

Initial experiments with two crystals showed that, with the b axis at approx 30 0 to the magnetic field, the effect on the photogene ration rate at 4.5 eV was insig­nificant compared with the current noise (mainly due to noise in the light source). There was no observable variation in the dark current, which was more than 2 orders of magnitude smaller than the photocurrent. These measurements were made with the deuterium lamp 4 m from the magnet and shielded by two concen­tric cast iron cylinders. The mains voltage was sta­bilized by two constant voltage transformers (600 and 100 W), before the deuterium lamp. The noise level in the light output was then 0.2% (peak to peak).

A more stable light source was a Bausch & Lomb tungsten filament lamp powered by a 6 V lead-acid bat­tery, independent of the mains. The photocurrent did not vary when a 2 kG permanent magnet was held near the tungsten filament, and the only other effects were transient effects caused by moving the permanent mag­net near the battery connections. The fields required for these transient effects were much larger than the stray magnetic field from the electromagnet and no similar transient effect was observed in using electro­magnet. The greater stability of the light source made possible more accurate measurements of the magnetic effect: a third crystal, at an applied field of 1.1 x 107

V m-1 showed a 0.07 (± O. 03)% decrease at 10 kG, inde­pendent of whether the angle between the b axis and the field was 0°, 15°, 30°, or 90 0

• At least four experi­ments were done at each angle and the results averaged.

III. RESULTS AND DISCUSSION

A. The dependence of current on electric field

For completeness, we summarize here criteria for distinguishing intrinsic from extrinsic photoconduction.

Extrinsic photoconduction in the case of anthracene crystals is generally caused by the reactions of singlet

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L. E. Lyons and K. A. Milne: Photogeneration in anthracene 'crystals 1477

excitons at the crystal surface, although a biexcitonic mechanism may be operative. The characteristics of photocurrents generated in this process are as follows:

(i) their spectral dependence follows the absorption coefficient20•21 ;

(it) they are dependent on the electrodes and the na­ture of the crystal surfacel ;

(iii) they are decreased if fluorescence quenchers are introduced into the crystal lattice21 ;

(iv) photon flux dependences are usually linear,21 but could be quadratic if a biexcitonic mechanism is opera­ti ve22;

(v) hole yields are usually much greater than electron yields. 20

In contrast, intrinsically generated photocurrents are

(a) independent of or only very weakly related to the absorption coefficient (hence they are independent of polarization of the light23

);

(b) independent of electrodes and the nature of the surfacel;

(c) unable to be quenched by fluorescence quenchers incorporated in the crystal lattice21 ;

(d) not greater than linearly dependent on photon flux2l;

(e) approximately equal for hole and electron cur­rents. 6

Figure 1 shows that the use of polyvinyl alcohol on the back surface of the crystal, plus the use of 1. OM Na2SOs contacts, Significantly decreased the current. We believe this indicates that extrinsic charge injec­tion (e. g., photoinjection from aqueous O2) from the back surface was prevented, so that the observed cur­rent was intrinsic. This is supported by the following observations:

(i} The absolute steady state photoconduction yields reported here USing aqueous contacts agree in magni­tude and field dependence with those for blocking and aluminum contacts and measurements by pulse tech­niques (Fig. 1).

(ii) The action spectrum for photogene ration of free charge carriers at high fields was independent of the use of aqueous or blocking contacts (Fig. 2).

(iii} P hotoconduction yields for holes and electrons were equal at the highest fields.

(iv) The photocurrent was proportional to the photon flux at either high or low fields over the range investi­gated (5x1010 to 2x1012 photons cm-2s-l ).

(v) The magnitude of the photocurrent was independent of the polarization of the light.

The process of intrinsic photogene ration of free charge carriers may be divided into three steps:

(1) Absorption of light to form an excited crystal state of lifetime"" 10-1S s, involving the vibronic manifold

of an excited Singlet.

(ti) Formation of an ion pair with an efficiency cf>1 , the distance between the positive and negative charge being ro' The Onsager modellO assumes an initial ion pair, and calculates the probability of the carriers escaping geminate recombination. The close agree­ment of the results of Chance and Braun7 with the low field predictions of the Onsager model indicates that, in anthracene, the assumption of ion pair formation is justified.

(iii) Escape of the carriers, with an efficiency cf> •• ,

to form free carriers in the conduction band.

The experimentally observed photoconduction yield cf> (electrons flowing in the external circuit per absorbed photon) may now be written,

(1)

We assume that cf>1 is a function of photon energy and is independent of the electric field, and so, the field dependence of cf> is due to cf>.s.

We consider two theories to explain the dependence of cf> on electriC field. (Neither the Poole-Frenkel nor the Schottky model is consistent with the results in Fig. 1, and neither is considered further,. )

1. The theory of Knights and Davis2 4

The theory of Knights and Davis24 assumes a model (see their Fig. 15) where the electron-hole pair is formed by thermalization of the hot electron, via dif­fUSion and scattering, to a position on the potential curve, at a distance r o, dependent on field, from the positive charge. The theory24 leads to

( 1 ) bEI

/2

IOg10 cf> •• -1 +2.303kT

eEro e 2

2.303kT + (47T£ro)2.303kT -loglolie T r' (2)

where b=(e s/7T£)1/2; £=£r£o; £" relative dielectric constant; £0' permittivity of a vacuum (Fm-I ); E, electric fieW (V m- I

); k, Boltzmann's constant; T, ab­solute temperature (K); e, the magnitude of the elec­tronic charge (C); lie, an attempt to escape frequency (S-I); T" the recombination lifetime of the electron­hole pair (s).

A graph of 10g10[ (1/ cf>u) - 1] + bE l / 2/2. 303kT vs E should yield a straight line with a gradient of ero/ 2. 303kT if ro is independent of field. We plotted (Fig. 4)

current-field data for one crystal, according to Eq. (2), assuming, in turn, three different values of cf>1

spanning the range of possible values of cf>l' We as­sume £ r = 3. B. 25 As no curve was a straight line, we could not estimate ro uniquely. In general, the slope increased as E tended to zero.

The gradient of the theoretical curve is given by

_ ero ( e)( e ) (~) - 2. 303kT + E - 47TE:Y~ 2.303kT dE

(3)

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1478 L. E. Lyons and K. A. Milne: Photogeneration in anthracene crystals

FIG. 4. Test of the Knights and Davis field dependence theory; (a) <Pl=l; (b) <Pl=8 X lO-3; (c) <Pj=5.3xIO-3. €r=3.8; T=298 K.

if, as Knights and Davis say, Tris independent of E. From Knights and Davis' model, dro/dE, if not zero, is expected to be positive. If dro/dE is positive, then Eq. (3) shows that dY/dE is expected to decrease as E- O. This was not observed (see Fig. 4). Assuming, then, that Yo is independent of field, the best fit to a straight line was obtained with <PI = 1. The linear portion of the curve gave Yo = 1. 9 nm. The predicted current-field re lationship (assuming r 0 is 1. 9 nm, E,. = 3. 8, and <PI = 1) is shown in Fig. 5, compared with the experimental results.

The Knights and Davis model can be made to fit the experimental results only for E > 107 V m-1 ; i. e., over a smaller range than does the Onsager theory. For this reason, the Knights and Davis model is not favored as the model on which to explain the experimental current­fie ld relationship.

Knights and Davis discussed also the variation of Yo

with photon energy (see Sec. I1I.B).

2. The Onsager geminate recombination theory! 0

Chance and Braun7 examined the current-field rela­tionship of the intrinsic photogene ration rate up to 2. 5 X 106 V m- 1 • Below this field, the Onsager current­field relationship does not depend on ro, and so no in­formation on Yo can be obtained from the current-field relationship. Above 2. 5x 106 V m-1, the shape of the graph of <Pe. vs E depends on Yo, and so sufficiently ac­curate current-field measurements can yield informa­tion on Yo .

OnsagerlO calculated j(r, e), the probability that an ion pair of initial separation ro at ,!-n angle e with the downstream direction of the field will escape geminate recombination. Onsager derived

(4)

where 2q=e 2/41TEkT; and (3=eE/ZkT.

Following the treatment of Batt, Braun, and Hornig4,5

and Geacintov and Pope,6 we assume that the ion-pairs formed initially are oriented isotropically in space, and that the delta function

(5)

describes the distribution of the carrier pairs as afunc­tion of r. Then the total efficiency of escape of the carriers is

dJ es = S fry, e)g(y)dT = ~ { j(ro, e) sine de . (6)

Let prE) be the probability of escape at field E relative to that at zero field; then

p(E) = <Pes exp(- 2q/Yo)

1 S' ="2 0 exp[ - (3ro(l +cose)]

[ ~ (3"'+n(l + cose)m+n(2q)mrn ] .

x L.J , ( ) , 0 sme de . m, n=O In. m + 11 •

Writing the integrand as X sine, we calculated P by expanding X in a power series in (3, and then integrating with respect to e, term by term. For evaluation by computer,

~ " "-p (-llP(2q)iY "0-i X= 1 + L (3"(1 + cose)n L L (7)

"=1 p=o i=O P! (n - p) ! j !

It may be shown that

'" 2" n n n-p (_ 1)P(2q)iyno-i

P(E)=l + L: _(3_ L L (8) n=1 11 + 1 p=o i=O P! (n - p) ! j !

The computer program calculated26 each term in (3 n ,

neglecting terms less than 10-5•

For hv = 4.5 eV, Fig. 6 shows lOgI0P(E) as a function of El/2 and Yo (for Er= 3.8 19; T, 298 K), compared with the experimentally observed current-field relationship.

6

FIG. 5. Comparison of experimental results with Knights and Davis field dependence theory (see Footnote 18). -. - _ ex­perimental results, T= 298 K. -- calculated from theory, ru=1.9 nm; E r =3.8; T=298 K.

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L. E. Lyons and K. A. Milne: Photogeneration in anthracene crystals 1479

o 15 g

- 9'0

-10,0

2 3 4

•• 2.0 (nm)

0'0 o

5 §

FIG. 6. Variation of the relative probability, PtE), of elec­tron escape with electric field: -- calculated from On­sager's theory (see Footnote 9); T = 298 K; € r = 3.8; ro as shown. • experimental results; T = 298 K; hv = 4. 5 eV. (J,

current).

Above about 106 V m-1 good agreement27 is shown be­tween the shape of IOglO p(E) and logl0! curves, provided ro = 2. 4 ± 0.1 nm. Thus we have a method to determine the value of r a •

The experimental curve of Fig. 6 was fitted in the same way for Er = 4.528 and 3.2,7 and gave ro = 1. 6 ± 0.1 nm and 3.0 ± 0.1 nm, respectively. The value deter­mined for ro varies Significantly with the value taken for Er •

The dielectric constant of anthracene has been mea­sured as 4.528 in the c' direction but more recent mea­surements by Munn25 give Er , c' = 3. 8 and an average of the three principal values as 3.2. Munn's values agree with Nakada's29 optical measurements. In the Onsager theory, with its assumption of an isotropic di­electric constant, it is necessary to choose a value of Er • According to Bass, 30 at high fields the effect of the electric field on the separation is chiefly in the c' direction. In the present paper we have used various values in order to display the effect of varying Er , but prefer 3.8.

We conclude that the Onsager model is adequate to explain the results. The reason that the Poole­Frenkel, Schottky, and Knights and Davis models do not apply is that each fails to consider the effect of three-dimensional diffusion which greatly enhances the probability of carrier separation at lower fields, es­pecially in solid anthracene where the mean free path is of the order of a lattice spacing and the potential bar­rier maximum is a number of lattice spacings away. We have shown that at high fields the Knights and Davis model agrees with the Onsager model, as expected.

B. The dependence of r 0 on photon energy

The observed variation in the current-field relation­ship with photon energy was interpreted with the aid of

the Onsager model. Each curve was compared with the predicted Onsager current-field relationships, and ro determined for each photon energy. Figure 7 shows, for two crystals, three experimental curves spanning the range of ro's observed. ro was determined in this way for 16 photon energies between 3.8 and 5.0 eV, for Ey= 3. 2, 3.8, and 4. 5.

Figure 8 shows the observed dependence of ro on photon energy. The increase in r 0 as the energy de­creased from 3.9 eV was unexpected. The effect pos­sibly may be explained as a background surface photo­injection process or surface photogene ration followed by trapping and optical detrapping. This process probably also occurs at photon energies greater than 3.9 eV, but the onset of intrinsic photogene ration with a much larger yield swamps the background effect.

Between 3.9 and 4.2 eV, ro is effectively constant at a value we denote by r OA ' For Er= 3. 8, rOA =1.8 nm. Between 4.2 and 4.5 eV, a transition region is observed where ro becomes larger. Between 4.5 and 5.0 eV, ro is again constant at rOB, equal to 2.5 nm for Ey= 3. 8.

We cannot place much importance on the absolute values of r OA and rOB for two reasons: (i) the ro values are only a few lattice spaCings (in the c' direction the spacing is about 1 nm) and therefore the assumption of a dielectric continuum is questionable; (ii) we assumed g(r) is a delta function (spherically symmetric) indepen­dent of field. This is the Simplest approach and involves no variable parameter. Gaussian and exponential func-

i w-.... '" (.) en

> 0:

'" 0: !:: a:I 0: ~

1-1 0

<5 0 ....

-8'9

-9,0

- 9·1

-9,2

-9·3

-9'4

r $!

c.:I o ..J

FIG. 7. Variation of intrinsic photoconduction with electric field and photon energy: ! I experimental results, for two crystals, T=298 K; -- PtE) (relativeprobabilityofelectron escape, compared with that at zero field) calculated from On­sager's theory (see Footnote 10); (a) hv =4.1 eV; ro = 1.6 ±0.1 nm. (b)hv=4.5eV; ro=2.4±0.lnm. (c)hv=4.9 eV; ro =2.7±0.1 nm. T=298 K; €r=3.8.

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1480 L. E. Lyons and K. A. Milne: Photogeneration in anthracene crystals

3·0

i E 2·0 ..s

1·0

3'8 4·0 4·2 4·4 4·6 4·8

PHOTON ENERGY (eV)---+

FIG. 8. Var iation of ro with photon energy. t! ! O. 3 e V bandpass. i 2 2 0.1 eV bandpass. (a) tr=3.2, (b) Er =3.8, (c) Er =4.5.

tions as well as a nonspherically symmetric delta func­tion have been considered in relation to the temperature dependence of the intrinsic photoconductivity in anthra­cene9, but conclusive indication of a particular function was not observed.

The important result is that ro does not increase steadily as the photon energy (hlJ) increases. Instead, either r 0 = r OA (hlJ = 3. 9 to 4.2 eV) or r 0 = rOB (hlJ = 4. 5 to 5 eV) or else ro, presumably, is an average of r OA and rOB (hlJ = 4.2 to 4.5 eV). Each ion pair state we envisage as having its own characteristic energy, E A

for r OA and EB for rOB' The relation between EA (or E B) and the photon energies giving rise to the ion pair states remains to be considered.

Qualitative confirmation of this result is found in the work of Chance and Braun. 8

•9 They studied the tem­

perature dependence of the intrinsic photoconduction yield for hlJ from 4.1 to 6.2 eV. Between 4.4 and 5.2 eV, the measured activation energy was constant, cor­responding to a constant ro of 5 nm, if E: r = 3. 2 and a delta function is assumed for g(r); the activation en­ergy was again constant for hlJ between 5.4 and 6.2 eV, corresponding with a constant ro of 6.7 nm. From 4.4 to 4.1 eV (their limit) the activation energy increased monotonically and so ro decreased. Qualitatively, our results are similar to those of Chance and Braun from 4.1 to 5.0 eV, where the experiments overlapped. However, our value of rOB, which is (for E: r = 3. 2) 3.2 ± O. 2 nm, differs from their value of 5.0 nm. The source of the disagreement may lie in the fact that

Chance and Braun's value was determined from tem­perature dependence measurements, but our value was obtained at constant temperature.

Onsager did not consider any dependence of ro on hv. The Knights and Davis theory for the dependence of ro on photon energy assumes that the process of separation of the carriers, immediately after optical excitation, is diffusive and involves the loss of the ex­cess kinetic energy by phonon emission. Assuming E = 0, they obtained:

(Ego the bandgap; lJ p , a typical phonon frequency; D, the diffusion coefficient of the carrier).

(9)

We used our Yo values from the application of the Onsager model and plotted [hlJ+e 2/47TE:Yol vs Y~ (Fig. 9) . A phonon frequency of 1013 (± 40%) S-l (not unreasonable) was calculated from the slope of the least squares fit (correlation coefficient = O. 87) to the data, assuming, for example, that D is equal to the diffusion coefficient of carriers in the lowest conduction band (0.026 cm2 S-l).

The intercept gave Eg as 3.7 ± 0.3 eV, again reasonable. The gradient of Yo with respect to field,

dyo _ er

dE - 2hv;Yo/D+eR/47TE:Y~-eE (10)

was calculated at 4.5 eV, (where Yo= 2. 5 nm) and for E=107 Vm- 1 :

dYo/dE=2.7xlO-9 nm/(Vm-1)

showing that according to this interpretation, r 0 is essentially constant with field.

It is not clear what value of D should be used, or even if Knights and Davis' initial assumption of a dif­fusive mechanism is justified. The process of thermal­ization must involve transitions between the conduction bands, which would not be described by a diffusive mechanism. The narrowness in energy of the lowest

5·0

I >-..

.!! ... 4·5 I:; .. -.. .. + .,.. ..c:

j 6 7 8

FIG. 9. Test of Knights and Davis theory for variation of ru with photon energy. I 0.3 eV bandpass. 20.1 eV bandpass. Er =3.8.

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L. E. Lyons and K. A. Milne: Photogeneration in anthracene crystals 1481

..t 'e u '-0 e

",-

c:> .:.

'"

T -&.

0

IS 0 ....I

i -Q.

2 \!) 0 ....I

i .... 0-2

\!) 0 ....I

t "G-

o IS 9

1. ".: 0

.... -;::-<I

30

20

10

2

I(i)

0

-1

-2

-3

-1

-2

-3

-4~

-49

-50

-4

-5

0·2

0·1 3·0 '

(a)

(e) i I

~ I I I

I I I I

! 4'5 5·0

PHOTON ENERGY (eV) __

FIG. 10. Variation with photon energy of the following: (a) absorp­tion spectrum of (001) face of crystalline anthracene [L. B. Clark, J. Chem. Phys. 53, 4092 (1970)]; -- lib; ......... Ib; EO, molar extinction coefficient; (b) yield of ion pairs; r, one photon (<Pt); -e-e- two photon Gi), calculated from data of Footnote 26; EO T ~ 4. 5; (i) Systematic error in absolute value of <Pt; (c) yield of ion pairs; r, one photon (<Pt); -e-e- two photon (Tj), calculated from data of Footnote 26; EO T ~ 3. 8; (d) two-photon absorption cross-section (o{), (see Footnote 26). (e) photoconduction yield (<p); (f) magnetic field effect on prompt fluorescence (see Footnote 34). t::.F! F (%) is the percentage change in prompt fluorescence with application of a magnetic field of 4 kG lib axis at room temperature; arrow indicates twice the energy of a triplet exciton.

conduction band precludes the possibility of thermaliza­tion solely within this band. The quantities evaluated from the Knights and Davis model for carrier produc­tion at ro are consistent; however, the agreement of the graph in Fig. 9 with a straight line is poor and sup­ports doubts about the applicability of the diffusive mechanism for the process of thermalization of hot car­riers, especially over such a small number of lattice spacings. We regard the Knights and Davis model as a zeroth order approximation to the real situation, and derived parameters as very approximate. Knights and Davis theory predicts a steady variation of ro with hll, which is not observed. Instead ro changes discontinuous­ly with hI' (Fig. 8).

C. The efficiency 1/>1 of producing ion pairs as a function

of hv

USing the model of Sec. m. A, with ¢ •• given by On­sager theory, we have:

(11)

where P(E) = 1 when E = O. Therefore,

¢ ¢ CP •• (O) P(E) = P(E) exp(- 2q/ro) = cPI

(12)

Values of CPI as a function of photon energy, calculated from Eq. (12) for €r = 3. 8 and 4.5, are shown in Fig. 10, together with some results of others (see Sec. N).

The remarkable features of our cPI values are (i) the appearance of a peak at 4.2 ± 0.15 eV and (ii) the fairly high efficiency at the peak. It is interesting that a peak is observed in ¢I at all, since one would expect that any excited state above about 4.0 eV would have enough energy to form either ion pair. The observation of a peak in CPl could possibly be explained by another decay channel becoming operative above 4.2 eV. Figure 10(b) does not imply that CPl > 1 since there was a systematic experimental error, the size of which is shown in Fig. 10(b), in the determination of the absolute value of CPl' Consideration of the errors shows that ¢l is not neces­sarily greater than 1. In the case of Fig. 10(b) the ex­tremely large values of CPl (Le., ¢I >1) may result from the possibility that 4.5 may be too large a value for €T' The maximum value of ¢l occurs at a lower photon energy than the maximum value of ¢ (4.5 eV). This is because ¢ •• is greater at hv = 4.5 eV than at 4.2 eV. As the TOA state must be reached indirectly by relaxation from the upper vibronic singlet manifold, we cannot identify the photon energy at the peak of ¢l with the energy of the TOA ion pair. Nor can we associ­ate the observed threshold of 1>1 with the energy of the TOA state, for three reasons: (i) some excitation from vibrationally excited levels of the crystal electronic ground state must occur; (ii) the bandwidth used in de­termining CPl near its threshold was 0.3 eV; (iii) for hv less than 3.9 eV some other process predominates with high ro values, masking the observed threshold of production of the states with roA •

If process (iii) is assumed to predominate below the "true" threshold only and allowance is made for (i) and (ii) then we estimate the true threshold to lie at about 4. 1 eV, and this threshold can possibly be taken as the maximum energy EA of the ion pair state TOA' By sim­ple calculation it follows then that the maximum value of E B = E A + 0.06 = 4.1 6 eV, and the maximum energy of free carriers in this conduction band is EA + 0.21 = 4. 3 eV.

The relatively high value of CPl at its maximum indi­cates that relaxation to the ion pair must be fast, com­parably so with relaxation in the vibronic manifold.

IV. COMPARISON WITH OTHER WORK

Baessler and Killesreiter3l estimated thermalization distances in anthracene crystals from the dependence of the photocurrent on photon energy, making the assump­tion that ¢1 was independent of photon energy. The

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1482 L. E. Lyons and K. A. Milne: Photogeneration in anthracene crystals

present results show 1>1 to be grossly dependent on en­ergy, and therefore the determination31 of the band gap needs modification. Also, the data of Batt, Braun, and Hornig4• 5 which formed the starting point for the deduc­tions of Baessler and Killesreiter have been modified by Chance and Braun. 7

Bergman and Jortner32 measured the photoconduc­tivity in anthracene induced by two photon absorption. At zero field,

1)(hv) = l7(hv) exp[ - 2q/ro(hv)] ; (13)

l7(hv), yield of ion pairs; 1)(hv), experimental yield of free charge carriers; 2q = e2j 4rr€okT.

Comparison of Eq. (13) with Eq. (12) shows that 1)

and 17 correspond, respectively, with 1> and 1>1 of one­photon experiments. We calculated fj from Bergman and Jortner's values of 1) and our values of ro for €or

=4.5and3.8. The results are shown in Fig. 10, to­gether with the two-photon absorption cross section aX from the same publication. Good agreement is found between 1>1 and 17. Because of our treatment with ro, the structure in 1), ascribed by Bergman and Jortner to vibrational structure in the conduction band, has dis­appeared in 1j for €or = 4.5 and is greatly reduced for €or = 3.S. Bergman and Jortner concluded that the sharp peak in 1) at 4. OS eV signified a direct onset to the Con­duction band, and that 1) and 1j provide a reasonable measure of the weighted density of states in the con­duction band. On the basis of the Onsager model, how­ever, the state produced following absorption at 4.1 eV is the ion pair state with r 0 = r OA, and, since the varia­tion of r 0 with photon energy is more pronounced than assumed by Bergman and Jortner, 1j and ¢1' not 1) and 1>, reflect the contribution of the ionic state wave func­tions to the overall wave function. The magnitudes of 1>1 and 1j are large enough for one to expect reasonably intense optical absorption to that state, symmetry rules permitting. The one-photon absorption spectrum does not show a peak at 4.1 eV, consistent with the calcu­lated transition probabilities to ion pair states or to the conduction band being very small. SS-S5 The two-photon absorption cross section, S2 however, has a fairly in­tense peak at 4.13 eV, as yet not definitely explained (see Fig. 10) (see also Footnotes 36 and 37).

Experimentally, it is consistent to assign the peak in the two-photon absorption cross section at 4.13 eV to excitation to the ion pair state (ro=roA ). The fact that it is not seen in the two-photon absorption cross section of anthracene solutions38 supports its assignment to a transition to an ion pair state. Calculations39 have shown however that the cross-section for true two pho­ton absorption from the ground state to an ion pair state is very small. Perhaps some other multiphotonprocess may be postulated in order to reconcile experiment and theory.

In the region from 3.9 to 5 eV, consequences of pho­ton absorption other than charge carrier generation have been reported. Figure 10(f) shows the energy depen­dence of the magnetic field effect on the prompt fluores­cence, measured by ArnOld, 40 who interpreted the structure observed in terms of a theory of Swenberg,

Ratner, and Geacintov41 which had been proposed for the optically induced fiSSion of Singlet states in tetra­cene at 77 K. The broad explanation for the structure observed is that fission occurs of excited singlet states into vibrationally excited triplet-triplet pairs. Another possibility, discarded by Swenberg et a1. ,41 is the for­mation of charge-transfer excitons in a process which competes with the formation of fissionable Singlets. Furthermore, in anthracene, the magnetic field effect on the prompt electroluminescence42 shows an activa­tion energy of 0.2 eV. An activation energy for the process implies that a metastable state exists for long enough to gain SkT of thermal energy. Therefore, we choose to regard the initial state in the fission process as a single relatively stable fiSSionable singlet state which may be thermally activated to give a triplet-trip­let pair, and this pair may then either split into two triplets or return to the metastable Singlet from which it was formed. Thus, according to our model, the en­ergy dependence of the magnetic effect on the prompt fluorescence reflects the energy dependence of the Yield, 1>s> of the fissionable Single state.

Figure 10 shows that 1>s does not have the same en­ergy dependence as 1>1, and that, therefore, the lowest ion pair state is not the fissionable singlet state. Fur­thermore, the magnetic effect on prompt fluorescence (even if ion pairs are involved) is certainly not com­pletely determined by the efficiency of production of ion pairs, as is evidenced by the lack of close corre­lation between the spectral responses of t:.F/ F and $1'

In addition, the lack of a measured magnetic effect on C/J at 4.4 eV shows that the process inhibit~d by the mag­netic field in Arnold's experiment40 is not a process which leads to ion pair production, nor a process which Significantly competes with ion pair production.

The magnetic effect on prompt electroluminescence has been explained by Schwob and Williams42 in terms of charge transfer exciton fission, though without com­pelling evidence that the fissionable entity was actually the charge transfer exciton. We have shown that, dur­ing optical excitation, the fissionable entity is not the charge transfer exciton (or ion pair state), but some other singlet state. The fact that ion pairs are formed during recombination of electrons and holes is not suf­ficient evidence for the ion pair state being the fission­able entity. The magnetic variation of prompt electro­luminescence is the same in sign and magnitude as that of the prompt fluorescence, excited above 4.3 eV. Therefore we assume that the same fiSSionable singlet is involved in both experiments, and also that the path­way for relaxation of the ion pair state is independent of its mode of formation.

The question now arises as to the identity of the fis­sionable singlet state. Schwob and Williams42 calcu­lated the energy of the fissionable singlet as 3.46 (±O. 05) eV and its lifetime as 10-9 s, assuming all the absorbed energy relaxed via the fissionable Singlet. The energy of 3.46 eV is too low from our results for the energy of the ion pair state which therefore is not the fissionable singlet. A lifetime of 10-9 s is rather long compared to the lifetime with respect to vibra-

J. Chem. Phys., Vol. 65, No.4, 15 August 1976

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L. E. Lyons and K. A. Milne: Photogeneration in anthracene crystals 1483

tional relaxation and is consistent with the observation of an activation energy for fission. A singlet state in the energy range concerned is the 1 L/) (lB3:) state, the energy of which as yet is undetermined.

One point remains: Ewald and Durocher report43 the action spectrum of the delayed fluorescence in anthra­cene. Because the spectrum showed a peak at 4.4 eV, similar to the photocurrent action spectrum, they con­cluded that the delayed fluorescence was caused by elec­tron-hole (general) recombination and subsequent fis­sion to produce triplets. Although some of their de­layed fluorescence may well be associated with general recombination of electrons and holes, it is possible that the processes they observed included the forma­tion of fissionable Singlets. Perhaps it is Significant that the magnetic effect on delayed fluorescence rises to a maximum at 4.4 eV. The lack of close correlll­tion between the energy dependences of delayed fluores­cence and CPl indicates that the de layed fluorescence is not determined by the efficiency of ion pair production.

V. SUMMARY

(i) Anomalous measurements of the intrinsic photo­conductivity of anthracene with aqueous electrodes have been resolved by the use of 1M Nazg03 aqueous contacts. The values of absolute steady-state photoconduction yield reported here agree with those measured USing blocking or metal electrodes, and pulse techniques.

(ii) The electric field dependence of the photoconduc­tion yield was explained more precisely by the Onsager theory of geminate recombination than by the Knights and Davis field dependence theory. Comparison of the experimental results with Onsager's theory yielded values of ro as a function of photon energy.

(iii) The Knights and Davis theory of the energy de­pendence of ro did not explain the results.

(iv) Two ion pair states appear to be formed: be­tween 3.9 and 4.2 eV with an ro of 1. 8 nm and between 4.5 and 5.0 eV with an ro of 2.5 nmo In the region of 4.2 to 4.5 eV ro undergoes a smooth transition.

(v) The spectral dependence of CPl has a peak at 4.2 ±0.15 eV with a value of 0.1, agreeing with the spectral dependence of 1j (two photon ion pair yield). The two photon absorption crOss section a{ has a peak at 4. 1 eV which appears only in the crystal spectrum and not in the solution spectrum. Thus, experimentally, the peak in the two photon absorption spectrum may be as­signed to absorption to an ion pair state. However, this assignment contradicts theoretical calculations of af for a transition to an ion pair state. A suitable ex­planation of the high intensity of af at 4.1 eV is needed.

(vi) In the process of Singlet exciton fission into two triplet exictons, the lowest ion pair state is not the fissionable singlet.

(vii) Only a very small [0.07 (±0.03)%1 direction in­dependent magnetic field effect on the intrinsic photo­current was observed at a photon energy of 4.5 eVo

ACKNOWLEDGMENTS

The authors gratefully acknowledge discussion with Professor C. L. Braun, and prepublication details of the work of Chance and BraunQ provided by him. We also acknowledge discussion with Professor L. Bass and Dr. J. S. Bonham, and support from the Australian Research Grants Committee (to L. E. L.) and the Aus­tralian Government (to K. A. M.).

lG. Castro and J. F. Hornig, J. Chem. Phys. 42, 1459 (1965). 2D• M. Hanson, Crit. Rev. Solid State Sci. 1973, 2430 3N• Geacintov and M. Pope, J. Chem. Phys. 45, 3884 (1966). 4R . H. Batt, C. L. Braun, and J. F. Hornig, J. Chem. Phys.

49, 1967 (1968). SR. H. Batt, C. L. Braun, and J. F. Hornig, Appl. Opt. Suppl.

Electrophotog. 20 (1969). 6N. E. Geacintov and M. Pope, Proc. 3rd Int. Conf. Photo­

conduct., Stanford, 289 (1969). 7R. R. Chance and C. L. Braun, J. Chem. Phys. 59, 2269

(1973). BC. L. Braun, Session for Discussion, Seventh Molecular

Crystals Symposium, Nikko, Japan, September 1975. 9R. R. Chance and C. L. Braun, J. Chem. Phys. 64, 3573 (1976). 10 L . Onsager, Phys. Rev. 54, 554 (1938). IIF. Lipsett, Can. J. Phys. 35, 284 (1957). 12L. J. Warren, Ph.D. thesis (University of Queensland, 1969). 13J. S. Bonham, Ph.D. thesis (University of Queensland, 1973). 14F • D. Bloss, An Introduction to the Methods of Optical Crys-

tallography (Holt, Rinehart and Winston, New York, 1961). IS(a) Geacintov and Pope [J. Chem. Phys. 50, 814 (1969) and

also Footnote 3] used a polymer film to reduce the extrinsic injection from the back surface of the crystal. We found that the use of polyvinyl alcohol gave better results than the poly­mer they used. (b) We found also that the use of 1M N~S03 in the back electrode improved the reduction by the polyvinyl alcohol film of the back surface extrinsic injection (see also Sec. III. A. This was possibly because the sulfite removed oxygen and prevented carrier injection by a reaction of sin­glet and/or triplet excitons with oxygen. No electron pho­toinjection was observed with the aqueous 1M NaS03 solu­tions. (c) Reduction of the extrinsic injection was most efficient after the crystal had been in contact with the 1M Na2S03 for several days.

16C • L. Braun and J. F. Hornig, Proe. 4th Mol. Cryst. Symp. Enschede, The Netherlands, 121 (1968).

17W. Helfr ich, Physics and Chemistry of the Organic Solid State, edited by D. Fox, M. M. Labes, and A. Weissberger (Inter­science, New York, 1967), Vol. III, p. 1-

lBAccording to Bonham (private communication), jet: Un+l, where j is the current density, U the applied voltage and n < 1 is possible for space charge limited currents only if d(lnp,)/d(lnp) <1, where P, and p are the free and total car­rier denSities, respectively. This means that (p/kT) dE,/dp <1 where k is Boltzmann's constant, T is the absolute tem­perature, and E, is the Fermi level. But kT dp/ dE, = J{h(E) x [1 +exp(E,-E)/kTr1} dE<p where h(E) is the trap distribu­tion, i. e., (p/kT) dEli dp > 1. Thus d(logj)/ d(logU) 2: 2 for any trap distr ibution.

19L. E. Lyons and K. A. Milne (to be submitted). 2oB. J. Mulder, Philips Res. Rep. Suppl. 4 (1968). 21R. F. Chaiken and D. R. Kearns, J. Chem. Phys. 45, 3966

(1966). 22 M. Schott and J. Berrehar, Phys. Status Solidi B 59, 175

(1973). 23N. Geacintov and M. Pope, J. Chem. Phys. 47, 1194

(1967). 24J . C. Knights and E. A. Davis, J. Phys. Chem. Solids 35,

543 (1974). 25R. W. Munn, J. R. Nicholson, H. P. Schwob, and D. F.

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1484 L. E. Lyons and K. A. Milne: Photogeneration in anthracene crystals

Williams, J. Chem. Phys. 58, 3828 (1973). 26We also modified the program to calculate P(E) as done by

Geacintov and Pope (see Footnote 6). They retained all terms in the summation in Eq. (4) from n = 0 to 00 and rn = 0 to 6 only. The two methods agreed to five significant figures. For the parameters used by Geacintov and Pope (€r = 3.02, ro = 8.3 nm, T = 300 K), we confirmed that P = 0.83 for a field of 4x 107 V m-I.

27 The deviation from the predictions of the Onsager theory for E < 106 V m -I has been observed also by Chance and Braun 7

in pulse measurements, and shown to be explained by free elec­tron-trapped hole recombination. It is possible to explain our steady-state measurements similarly.

28G• R. Johnston and L. E. Lyons, Phys. Status Solidi 37, K75 (1970).

29I. Nakada, J. Phys. Soc. Jpn. 17, 113 (1962). 30L. Bass, Trans. Faraday Soc. 64, 2153 (1968). 3tH. Baessler and H. Killesreiter, Mol. Cryst. Liq. Cryst.

24, 21 (1973). 32A. BergmanandJ. Jortner, Phys. Rev. B9, 4560 (1974). 33L . E. Lyons, J. Chem. Soc. 1957, 5001-5007. 34R . S. Berry, J. Jortner, J. C. Mackie, E. S. Pysh, and

S. A. Rice, J. Chem. Phys. 42, 1535 (1965). 35S. D. Druger, Chem. Phys. Lett. 17, 603 (1972). 36 F . C. Strome, Phys. Rev. Lett. 20, 3 (1968). 37R . G. Kepler, Phys. Rev. B 9, 4468 (1974). 3BA . Bergman and J. Jortner, Chem. Phys. Lett. 15, 309

(1972). 39I • Webman and J. Jortner, J. Chem. Phys. 50, 2706 (1969). 40S. Arnold, J. Chern. Phys. 61, 431 (1974). 4tC. E. Swenberg, M. A. Ratner, and N. E. Geacintov, J.

Chern. Phys. 60, 2152 (1974). 42 H. P. Schwob and D. F. Williams, J. Chern. Phys. 58,

1542 (1973). 43M. Ewald and G. Durocher, Chern. Phys. Lett. 12, 385

(1971).

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