Optical nuclear polarization as a consequence of the non-crossing rule (level-anti-crossing)

Chemical Physics 21 (1977) 289-299 0 North*HollandPublishingCompany

OPTICAL NUCLEAR POLARIZATION AS A CONSEQUENCE OF THE NON-CROSSING RULE (LEVEL-ANTI-CROSSING) 11.The influence of electronic relaxation

D. STEHLIK * and J.P. COLPA ** MaxPlanck Institute, Department of hfolecuhr Pl:ysics. 6900 Heidelberg, FRG Received 4 October

1976

The theory for Optical Nuclear Polarization (ONP) as a consequence of Lcvcl-AntiCrossing (LAC) as outlined in rhc first part of this series, is extended by considering the influence of relaxation between the electron spin states 7’,. T,, and Tr. Unselective relaxation as defined by equal rates of relaxation among the spin states, T,,. in general reduces the nuclear polarization. In addition, it broadens the shape of the ONP as a function of the magnetic field. Similar destructive effects are found for selective relaxation with the exception of one specific type of selectiveness. It will be referred to as constructive relaxation, because it is even able to enhance ONP by LAC together with a more pronounced broadening of the field dependence. For instance, when a polarizing field is taken in the z direction, a large rehsation rate l~‘~,, (relaxation between rx and Ty) enhances the effect. Large rates Wxz and rYyz would reduce the ONP. The constructive relaxation is treated analytically by perturbation theory. From the experimental ONP results one has to conclude that often one of the three relasation rates is considerably larger than the other two.

1. Introduction

ONP has been observed in several pure and doped molecular crystals. Although more and more data could be explained by a general mechanism developed in refs. [Z-4] as reviewed in [5,6], the most important predictions of the mechanism concern the regions of level crossing among the electronic triplet spin states [2,6]. Recently they received renewed attention because experimental evidence for ONP in the levelcrossing region became available. ONP as a consequence of level-anticrossing can be treated analytically as shown in the preceding paper [l] confirming and elucidating the numerical results of the general model [4,6]. The model turns out to explain in principle the most prominent ONP-data in doped fluorene crystals assuming the existence of triplet state complexes among host and guest molecules. * Present address: Freie Universitat Berlin, Fachbereich (PB 20), lnstitut Atom und Pestkorperphysik WE 1B. Klinigm-Luise-Strasse 343.1 Berlin 33. l * On sabbatical leave from Department of Chemistry, Queen’sUniversity, (Ontario). Canada. -- Kinnston _

Physik

For the following it might be helpful to summarize briefly the essential elements of the ONP by levelanti-crossing [I]. Let 3s consider a molecular triplet state subject to a magnetic field of suitable strength and orientation such that two of its electronic spin states cross each other (level crossing). The hyperfine coupling splits the electronic spin states, in addition it couples them, causing a level-anti-crossing (LAC) involving selected pairs of repelling hyperfine levels. Each pair of levels is characterized by its own LACcenterfield and by its anti-crossing range. The coupling of states connected with each LAC involves also a redistribution of the population of the states Together with the selective population and decay of the electronic triplet spin states this results in a selective return to the nuclear sublevels of the ground state after an optical excitation cycle through the electronic triplet state. The nuclear polarizations contributed by each pair of anti-crossing hyperfine levels are mutuully opposite in sign. However, they do not cancel each other in general because they are characterised by their individual LA&enters and ranges. As a result, the typical field dependence of the ONP in a level ctoss-

290

D. Stehlih-, J.P. Colpa/Oh’P as a consequence of the nomcrossirg rule. II

ing region is the sum of two (or more) contribtitions of opposite sign, different width and different center fields [I]. Hence, the ONP in he level-crossing region is based on the population redistribution associated with level-anti-crossing induced by hyperfine coupling. The question arises then: how is this process influenced by relaxation which is known to affect populations as well. Generally. relaxation js cspected to destroy the nuclear polarisation created by the optical pumping; process, since it tries to reestablish thermal equilibrium, i.e.. ;I Boltzmann distribution over tile spin states (which can be approximated by equal populations for our purposes). Hence, relaxation is predicted to reduce ONP. However, this destructive effect of spin retaxalion might change to a constructive innuence on ONP (see 111) if the relaxation rates among the three electronic spin states are highly selective, i.e., very unequal. Although little has been said about the mechanism of selective electronic relasation in the excited triplet state, the fact has been established experimentally in many examples, particularly by ODMR-measurements in zero-field, refs. [7,8] giving recent examplcs. In addition, selective relaxation rates are espectcd since the corresponding transition matrix elements are subject to symmetry selection rules just as the electronic intersystem crossing rates. In this paper we deal mainly with the influence ofselecrive electronic relaxation on the ONP in the level.crossing re@on. It turns out that essential features of the experimentally observed field dependence can be understood this way. As a result, the remarkable sensitivity of ONP to selective relaxation offers an alternative and independent way to prove its existence and to determine its properties. In the present conte:;t the relaxation mechanism itself is of no importance. It is sufficient to start from any relaxation model with the following general features: (I) The interaction responsible for the reiaxation is likelyto be of predominantly electronic origin and this will be assumed here. (2) The three relaxation rates among the zero-field triplet spin states can be different, i.e., selective’relaxation is assumed. All relaxation rates in the presence of a magnetic field will be expressed as linear combjnation of the “zero-field rates”. The influence of such a relaxation on ONP will be

investigated following three different approaches. Firstly, a short section is used to develop a pictorial argument. Secondly, we extend the analytical treatment given in the preceding paper, which is restricted, however, to the high-field approximation. Finally, the numerical results obtained with the general model given in [4] are discussed and interpreted.

2. Qualitative approach to the relaxation influence WCstart with reference to section 4 in the preceding paper [ 11.There, a pictorial interpretation was given for the ONP in the level-crossing region. The essential arguments may be summarized as follows: Level-anti-crossing (LAC) occurs between pairs of hyperfine levels belonging to the following subspaces, I-?and 11~:respectively, as given in table 1. The LACpairs are characterized by their different centers H& and fIFic given in eq. (4) of [I]. At magnetic field HZ = HpAc, the states $4 and G5 are linear combinations of the pure product spin states with equal weight, hence they contribute equally to the repopulation of the ground state nuclear sublevels after an optical excitation cycle snd do not contribute to the ONP. The same is true for the high energy states $,

and G7_:In the high-field approximation they have the same electron spin character and equal population. Therefore, only the states $3 and q6 remain to be considered. They are different in both the electron and nuclear spin character as long as the field is sufficiently far away from their LACcenter at H&_ Consequently, a selective population or decay of the electron spin states results in a selective repopulation of the nuclear sublevels in the ground state. In the example used we assumed a preferential population of the state ti3 * [-)0(n) as a result of the selective population of the I+) and I--) electron spin states. Hence the nuclear p(n) spin sublevel wilt be overpopulated in the ground state representing a negative ONP. In general, one would argue that relaxation will tend to remove all population differences and thus reduce the ONP. However, this is not true for particular relaxation processes as we would like to point out now. We consider the special case that electronic rel‘axation occurs preferentiaIIy between the TX and TV electronic zero-field states, with a rate rVXY,connecting the I+) and I-) high-field states. By inspection of table 1

D. Stehlik, LP. ColpaJONPm

a comquence

Table I Basis Iunctions used for the subspaces Iz and IJz for the description of High energy state

It

1+)12(n)= -(l/JZ)(Tx

112

l+>Wi) = -CLiJz)(T~ f iT$P(n)=&

$2 * ($3, $6).

291

level-anti-crossing LAC pair

Subspace

f i$Ialn)

one realizes that this sort of relaxation can induce relaxation transitions only within the subspaces

k:

the

of thenon~crossing rule. Ii

171

At Hz = HFAc for the above example, the relaxation transition (1) does not alter the equal population of the states e4 and +5 which still do not contribute to the ONP. However, the population of the preferentially populated state $1 witl be reduced by relaxation of type (I). Since the field has been assumed to be far from the LAC center of subspace Ilz, relaxation (2) becomes $2 * $3, but these states have the same population anyway. Thus, another population difference NUl) < Wlii2) is created. The excess population in $2 will add another contribution to the negative ONP discussed before which results from the decay of state $3. Hence, we conclude an enhanced ONP as compared to the case without relaxation. It will become clear in the next section that this particular feature of the selected relaxation rate W,.. stems from the fact that it induces relaxation only within the subspaces, given in table 1. The other electronic relaxation rates connect the two subspaces and thus destroy the population differences found to be responsible for the ONP. They will turn out to be destnrctive with respect to the ONP while the relaxation rate W,.. (connecting the field dependent electronic spin states) is corzsrn&ve. Another important feature of the relaxation can be recognized within the qualitative discussion given here. We remember that the mixing of the pure product spin functions near the LAC-region,yccurs within certain ranges u (see eq. (6) of (11) which are generally different for the two subspaces. Consider a slight mixing with respect to the states fi4 and $Q in the wings of the LAC-region of subspace Iz. If the relaxation

= $1

i-1 a(n) = (1/~%3”~ - ir,l C&I), T&n) ; combin: +q 44 and $5 I-)P(n) = (l/J-)(Tx - iTy) P(n), T2c4n) ; combine to @s snd q6

transition (I) is sufficient& strong it will equalize the populations on the states $L, G4 and $5 although the mixing might be small. This population redistribution was argued above to cause ONP a~H&C_ Now we realize that with increasing relaxation rate 5U_YY, the popu~alion change is carried further and further into the wings of the LAGregion. Thus the electronic relaxation is predicted to cause a broadening in the field dependence of the ONP which increases with increasing relaxation rate. Furthermore. the broadening feature will appear more pronounced on the ONF contribution from the subspace with the larger range. With this qualitative discussion we hope to give the reader a more physical idea of the results the following quantitative treatments aim at.

3. Analyticai treatment of the influence of selective relaxation on ONP by UK We outline in this section how quantitative results can be obtained for the ent~an~en~ent by selective relaxation of ONP in the LAC region. The treatment will be based on the perturbation theory approach developed in paper I and on the general model for ONP in ref. [4]. The main purpose of this section is to formulate quantitatively the expectations expreaed in the previous section. We reconsider the case treated in the sections 2 and 3 of paper 1. We take the particular case of a polarizing field Hz parallel to the z axis and we assume that the higWield approximation is adequate. As mentioned in the preceding section, the relaxation component JyrYconnects here only states which belong to the same subspace, either It or 11~. We consider WX,= WYz= 0; Wx,, + 0. The condition ru,, = I$ = 0 has as a consequence that relaxation between the states @l, $J~, ti5 on the one hand and $2, +3, G6 on the other hand are impossible; one sees

this easily by inspecting the functions in table 1. In section 4.1 of ref. [4I we defined the relaxation rates ‘Vii according to our model. This definition to-

gether with the definition of the coefficients in section 3 and in particular the relations (10) and (11) of paper I gives the rates summarized in table 2. We take further a case for which the population of the triplet state is selective but the decay to the ground state is non selective. In the notation used in ref. [4] and paper 1 this leads to k, = kY = k3 = k/3 _

(3)

Formula (3) together with table 5 and formula (11) in paper I give then: k,=k,=X_,=X_,=~,=k,=k/3,

We can consider the (kN~)‘s as unknowns and lVij/k as given. We can also take the k and the Ni’s together in (5). This means that we can set conveniently k = 1 so that the overall decay rate k is chosen as a unit rate. ~Vijhas to be expressed in this unit rate, as outlined in connection with eq. (23) in ref. [4]. It is jntuitively obvious that “ii/k is the important parameter. Only when the relaxation is fast compared to the decay rate is it possible to redistribute the occupation els during the lifetime of the triplet state.

of the lev-

In general (7) gives for an electronic triplet coupled to one proton six coupled equations. When only A 0, as we assume, one gets two sets of three ‘“xy 7coupled equations, one for each subspace. For space I2 one gets:

(4)

This means, in words, that the decay constants of all six sublevels are equal. In paper I we derived a general expression (eq. (15))

P, = N, (; t WI4 + fV15) --iv, WI4 - Ns WI5 , P4=-N1W14+h$(:+~V14+1V~‘45)-NgW45,

(8)

for the polarization rare @T(O) under the assumption that the high-field approximation is valid. Substitu-

Ps = -A’, W,5 - Iv, WGs + Ns (4 + WI5 + Wd5)

tion of (3) in this formula gives:

From (8) follows that P, + P4 + Ps = i (NI + N4 + Ns) From the expressions for the population ratesPi as given in paper I (table 5) one finds, using also (11) in paper I:

FjT(0) =fk[Arl - N, +N,{lC;~\2 +A’s(lcl,12-Ic,,l

- IC+,l’)

+ 2)-N~(lCy-

q

PI +P&j+Ps = I/2, _- h’6(ICp12 - ICz;r12)J The occupation numbersNi in (5) with the rate equation:

(5) can

be calculated

and hence N1+N4+N5=3/2.

(9)

One can get explicit expressions for Nl, N4 and N5 Pi=kiNi

+C

‘Vii(Ni- Ni),

i= 1,...,6.

(6)

by solving three equations, most conveniently (9) and

(See formula (5) in ref. [4] .) Substitution of (4) in (6) and some minor rearrangement gives

two from (8). For N2, NJ and N6 one can get expressions in a similar way as for Nt, N4 and Ng. Substitution of these expressions in (5) gives after rearranging terms:

j+i

Pi = f (J-N,)

+ [z, (h’ij’iilk)[(kfii) -

(kNi)l>

i= 1) . ... 6. (7)

D. Stehh’k, J.P. Colpo/0NP~~sa comequence of rhe non-cmrsisg rule. II

--

LJl(2 f 9JVxy + 9JV$) 2 + 6Wv + 3L,(2JU,,

+ 9JV&,)) ’

(10)

When JYXv= 0, (10) becomes $08

= (2P, - PX - P,)WJ - &I) 9

(10

which is identical to (17) in paper 1. When LJ $0 (in the WC region around HFAC and IV-t=, one finds for the first term in the square brackets the value l/3. The lorentzian as defined in (10) in paper I has a maximum value l/4. This increase in the maximum value is related to the fact mentione’d in the preceding section that the relaxation equalizes the population of the states 1,4 and 5 whereas without relaxation the effect of the LAC is on 4 and 5 only. One easily checks that for large W the two terms within large parentheses get half their maximum value (l/6) when LI = 219JVXY

or

LI* = 2/9WXY.

The field strength for which we et half the maximum $ LJsmgthe detiniONP intensity is denoted by AHl,2. tion ofL, in(l0) in paper I, we get

It follows that aHf,* = laJ,l~,

with WX,,5 1 ,

(13

and hence, the half width becomes proportional to 6, There are, of course, analogous formulae for the peak at !#ic. There is a certain formal analogy between (10) and (16) in paper I. This is most easily seen by rewriting the first term in large parentheses in (10) as

For Wxv> 1 this term becomes approximately

29;

We have to compare 2/3Jq_, with 2X-:(/$ + ki) in (I 6) of paper I. Large JvXYgives broadening. By analogy one expects then that small tiZ(ki t ki) gives also a broadening without relaxation. Numerical calculations show that broadening by the decay selectivity occurs when either k: < I or kl+ k$ <
4. Numerical results While the analytical treatment given before allows certainly a deeper physical insight into the relaxation influence on the ONP, it can be worked out with sufficient simplicity only within the high-field approximation which is well applicabIe to the level.crossing situation on the upfield side (H IIz). Concerning the second level crossing case as well as the non-crossing situation (H IIx,y), the only practical way is the use of the general model as given in ref. [4]. In this SKtion we would like to present some numerical results in order to demonstrate the major relaxation effects on the ONP-field dependence referring back as much as possible to the physical concepts developed in the preceding sections. The use of the general ONP.mechanism as summarized in [4] requires the following input: (a) The parameters characterizing the total spin hamiltonian of the excited triplet state. (b) The rate constants governing the rate equations of the triplet state sublevels. As outlined before and in [4J, it is necessary and convenient to consider only relative rates for both, the population and the decay of the triplet sublevels. Then the relaxation rates have to be taken in units of k, the absolute decay rate defined in eq. (3). (c) The relaxation model, whjch is specified now as describedin section 4.1 of [4]. Only the purely electronic relaxation rates JVxy/k; JVxZ/k; Jv,,;,/k among the zero field triplet states appear as independent input values. The parameters used in the numerical examples are

294

D. Srehlik. J.P. Colpa/ONP0s 0 consequence of the nomcrorsing rule. II

given with the figure captions. They are intended to reproduce the experimental ONP-results observed in

doped fluorene crystals (see fig. 2 of part 111,[lo]). For the moment, however, they can be taken as an arbitrary set of parameters with the following general properties: (I) D > 0, E < 0,i.e., tile two level-crossing situations are H IIz and H IIx; H liy representing the noncrossing case. (3) IDI 5 [El. In that case, the high-field approximation holds to a very high degree of approximation for the uplield level.crossing situation H IIz. (3) The hyperfine tensor is assumed to have the same principal axis system as the zero-field splitting tensor and only one nuclear spin I = I/?- couples to the electron spin S = 1. 4.1. Tire influence of lVsy for rhe polarizing field H IIz

It has been pointed out in sections 2 and 3 that the electronic relaxation rate lvXYis predicted to be constructive for the ONP in the case H IIz. This relaxation connects those electronic spin states in an applied magnetic field. It is responsible for the IAml = 2 relaxation processes in high field. Fig. 1 presents the polarisation rate per created triplet state &(O) as a function of the polarizing field Hz for three values of the relative electronic relaxation rate rv,,/x- = 0,30, 100. The curves are calculated using the general mechanism of ref. [4], however, they are identical to the results obtained from eq. (10) with an accuracy ofbetter than 1% except for the region close to HZ = 0 where the perturbation treatment is bound to fail and only the accurate diagonalizationof the spin hamiltonian renders an ONP approaching zero with decreasing magnetic field strength. Inspection of fig. 1 renders the following genkral features: (a) The increase of the relaxation rate lvxv results in a substantial broadening of the ONP on the field scale retaining the overall amplitude of the polarisation. We call this the constructive relaxation effect. (b) The narrow positive polarization peak in the curve without relaxation is reduced, With fast relaxation it becomes finally a mar;imLm near zero polariza-

Fig. 1. Initial polarisation rate&(O) per created triplet state as a function of the polarizing magnetic field HP oriented along the z.asis of the zero-field splitting tensor. All curves are calculated using the numerical model described in the tex with the following parameter set: D/h q 394.8 hlHz, E/II = -15.3 MHz; A.& = -7 hlHz, ,.lyy//r = -21 hIHz, A,,//! = -14 h1H.z; px = 0.05. p = 0.9G,pz = 0.05; k_;;:=k’=kY=1/3. Th~foll~wing sets of selective electronic relaxation rates have been used: (a): W&c = W&k = Wyz/k = 0. (b): Wxv/k = 30, L&/k = 1$/k = 0. (c): Wxy//c= 100, WrZ/k = Q/k = 0.

tion. In addition, it appears to be more and more central within the broad negative region of ONP. Note thal the field position of this ONP-maximum is hard. ly shifted at all with increasing relaxation. It defines rather accurately the LAC-field H&J = Htc - +A,,/gfl and thus provides a good reference point for the determination of the level-crossing field Hfc. (c) The broadening of the negative and positive ONP.contributions can indeed be characterized by a respective width proportional to the square root of the relaxation rate as derived in eq. (12). It is this broadening which makes the narrow ONP-contribution to appear more and more symmetric near the center of the broad ONP-contribution since the width becomes large as compared to the splitting between the centers Hk& as given in eqs. (4) of ref. [l]. For a comparison with the experimental results it might be useful to summarize again those features in the field dependence of the nuclear polarisation rate that reflect the influence of the constructive relaxation:

D. Stehlik. I.P. COI~Q/ONP~S a conseqwnce offhe non-crossing rule. II

295

- broadening without reduction of the overall ONP; -the narrow ONP peak within a broad region of opposite sign drops with its extreme v&e to the zero level. One should add that the relaxation obscures a useful property of the result without relaxation. For Key = 0 the narrow positive ONP-peak clearly sits on the downfield side of the broad negative ONP contribution. Hence, the inequality (18a) of [I],

is easily assigned to the field dependence. One can extract a fairly recise value for the splitting element: lH?i; - $“LL&l = IAzZl/g/L Increasing relaxation rate fVxu makes it more and more difficult to recognize this feature. On the other hand, one realizes that the ONP field-dependence as well as any other indicator of the population redistribution in the LAGregion is a very sensitive detector of a selective relaxation process. In this context we emphasize that the ONP result with the magnetic field If 11g oriented along one or the principal axes p =x,y, 2 will be sensitive to a predominant relaxation rate WS,pSr with p f p’, .u”. This generalized statement has been noticed already in the case of the ONP in pure phenazine crystals [4] which represents the rlon-crossing case in good approximation. ft will also become more apparent when we discuss now the relaxation effects on the field dependence of the ONP for all three major orientations of the magnetic field.

4.2. Gelzeral Bljluerxe versus destnrcrive

of the relnxafion. Corrstnrctise

rel~atio~i

Fig. 2 shows the calculated ONP as a function of the field strength for all main orientations. The broken curves are taken from fig. 2 of [I] in order to compare with the case of no relaxation. The selectivity of the relaxation has been chosen to be constructive for the levelcrossing case H 11 x, corresponding to a predominant rate W,&. Let us consider first the influ. ence of this relaxation rate on the field dependence H Iiz discussed above. The bottom part of fig. 1 demonstrates the following properties: (a) In the first place, the relaxation rate WYrcauses a substantial reduction of ONP. Note that the scale of the solid curve with relaxation has to be increased by more than two orders of magnitude when the polar-

Fig. 2. Initial ONP rate @T(O)per created triplet state 3~3. function of the polarizing field Hp oriented along ~hcthree principal axes oF the zero4eld splitling tensor. The numerical calculation has been carried out with the parameters given in the caption of fig. 1, except for the relaxation rates. The broken curves reproduce the result without relasation given in iig. 3 of paper I. The solid curves are calculated with the following set of relaxation rates, all other parameters remaining unchanged. W&,/k = 0, IV&k = 0, Tr>Jk = 30. Note the different scales of&(O) in the bottom figurefor the broken curve (right sulc) and the solid ctirve (left scale).

isation extremes of both curves are normalized to about the same level for the sake of the representation. (b) The broadening due to this destructive relaxation is less pronounced and different for the positive and negative ONP contribution as compared to the

D. Stelrlik, J.P. Colpa/ONPas a consequence of the nonxrossing rule. II

296

case with constructive relaxation. (See fig. 1.) (c) The destructive rekation retains a substantial. positive ONP.peak. Correspondingly, the position of the posilive peak in the dowWield wing of the broad negative ONP-contribution is still clearly visible as compared to the effect of the constructive relaxation in fig. 1. Hence, even disregarding the polarisation amplia lude, the shape of the ONl’ field dependence exhibits typical properties allowing a clear distinction of a destruclive and a constructive relaxation rate described before. Turning now to Ihe second levelxrossing case H Iix (middle part of fig. 2) we find again the essential features of lhe influence of constructive relaxation as analyzed irk section 4.1 for the other level crossing case H IIz. However, the properties (a-c) given there are modified significantly due to the near-zero&Id region: (a) The broadening by the constructive relaxation W,,Zis quite obvious in the upfield region of the levelcrossing field ff&

= 41 G. In the downfield

regon

the same broadening combined with the requirements of a polarization approaching zero in zero-field results in a strong reduction

of the polnrisation

rate near

zero-field. (b) The narrow ONP-contribution associated with a significant negative ONF peak in the case without relaxation is again repfaced by.an ONP extremum at the zero.level. This feature is therefore common to both level-crossing situations, although the negative ONP contribution in fig. 2 (H IIx, It;lz f 0) becomes only a small bump on the downfield slope of the broad positive contribution. Again, this narrow ONP extremum provides the best reference point for the determination’of the LAC-fieId from the ONP-field dependence. (c) As pointed out in section 5 of [I] the concept of ranges assigned to the positive and negative ONPcontribution has to be handled with care outside the high-field approximation. Nevertheless, the overall broadening of the field dependence is still fairly well characterized by a proportionality to the square root of the relaxation rate, for WYZS 1. In ref. [I] it has been shown that the relative position of the centerfields of the broad and narrow ONPcontribution are important for the determination of the hyperfine tensor elements, in particular their signs.

For H )IX, the assignment of an inequality of type (25) in [ I] becomes rather difficult. The choice of the parameters in fig. 2 renders

This feature is stjll apparent in the doteed curve of without relaxation. The narrow negative fig. 2 (H Ilx) ONP-contribution is still on the upfield slope of the broad contribution although much less pronounced in comparison to the case H IIz as 3 consequence of the non-high-field approximation. With relaxation,

the above inequalily seems to be reversed. Nevertheless, the narrow ON?-contribution remains centered upfield from the levelcrossing field I&, but due to the field-dependent electron spin functions near zeros field, the broad OW-contribution is altered in shape so much that it appears to have its center at a higher field. Therefore, special care should be applied in the analysis of the corresponding ONP-curves. However, we see that the changes of the ONP-field dependence are much more pronounced when constructive reIaxation affects the low-field level-crossing situation as compared to the level crossing within the high-field approximation. Consequently, the influence of constructive relaxation can be detected with even higher sensitivity in the former case. Finally, we turn to the results for the field oriented along the third principal direction, the non-crossing case H Ily (see top part of fig. 2). In this case, the relaxstion WYZcauses relatively small changes. Both, the amplitude and the shape of the field dependence, are not much altered, althou& the distribution of positive and negative ONP-regions is changed significantly. The absolute values of the polarization rate are found to be comparable in the example of fig. 2 for

the orientations H Ily and H II z.In contrast, the polarisation rates reached for H IIx are more than two orders of magnitude higher as a consequence of the selective relaxation which is constructive for this case, but destructive in the others. However, this enormous difference in the polarization level reduces if the selectivity of the relaxation rates is decreased. Unselective relaxation (WXY= IV,, = I+‘& reduces the absolute polarization in all cases to the same level even though the polarization rates are highly different without relaxation.

D. Stehlik, J.P. ColpafONPas u consequenceof the rim-crossbg de. 11

297

4.3. Determination of the hyperfine tensor elemerm

described above for H IIZ, now the field dependence remains sensitive to the size of the hyperfine coupling,

In principle, the ONP field dependence can serve as an ahernative method for the determination of the hyperfine tensor, which might be of particular interest when standard maaetic resonance methods like EPR or ODMR fail due to the lack of signal or resolu. tion. The analysis of the ONP in the Iwsl crossing re-

even in the presence of fast constructiverelaxation. Finally, it should be pointed out, that the field dependence for the third, the non-crossing case, adds information which might be crucial to distinguish between different alternatives for the assignment of the hype&e tensor.

gion givenin [11 has shown that the hyperfine tensor cIementscan be read directly (a) from the splitting of the L.AC.fieIdsH$$ (see (4) and (25) of [I I), and (b) from the ranges uI 1l of the positive and negative ONP-contributions (sek (6) and (25) of [I)). However, we have seen in this paper how the determination of both, the LAC-fields and the ranges, can be limited in accuracy by the influence of relaxation. Therefore, we have carried out a set of numerical calculations in order to demonstrate which properties of the ONPfield dependence are most sensitive to the individual hyperfine tensor elements. We summarizebriefly the most important aspects, without showing the calculated field dependencies: (I) In the case of destructive relaxation (see N Iiz in fig. 2) both the splitting of the LAGfields and the ranges can still be extracted from the field dependence, however, with some reduced accuracy due to the additional open parameter, the relaxation rate. We would like to emphasize that the information about the signs of the hypertine tensor elements as contained in the inequalities (18) of [l] remain unaffected by the destructive relaxation. (2) Strong constructive relaxation alters the ONPfieId dependence such that it becomes more difficult to distinguish between effects due to changes of the hyperfine tensor, and changes of the relaxation rate, however, only for H IIz. For the second level-crossing case (H 11 x for E < 0) the field dependence of the ONPremainssensitive to both, the splitting of the

L4C.fields and the rangeseven when fast constructive relaxation is allowed.This is due to strongly field dependent electron spin functions near zero field. The numerical results show that the ratio of the ONP maxima (see fig. 2, HIlx), the one on the upfield side with respect to the one on the downfield side of H& , increases rapidly with increasing A,. This is quite plausibie since increasing A, shifts the downfield LAC-field closer to the zero-field where the ONP becomes increasingly reduced. In contrast to the situation

5. Comparisonwith experimentalresults

The most significantONP results so far have been ,observed in doped fluorene crystals at room temperature and in rather low fields (see refs. in [I]). Their explanation has been delayed for so long because it implies the postulation of peculiar triplet states of complexes among guest and host molecules, the existence of which has been proven only recently by ESR measurements [9] and can be concluded independently from the ONP results [6! 101. The zero field splitting tensor is characterized by unusually small D and Evalues and a principal axis system, termed x*,y*, a* which does not coincide with the molecular axes of nei ther the guest nor tfle host molecules except for the long in-plane ads x I\x* IIc which happens to be along the crystalline c-axis in all cases. These properties can be analyzed, from the ESR-results [‘3] as well as from the ONP as a function of the polarizing field oriented along the three main axes. In ref. [l] it has been shown that the field dependence of the ONP can be understood in principle if ii is assigned to the level-anti-crossing regions of the triplet state complexes induced by their electron-.proton hyperfine coupling. Althou& a detailed understanding of the upfield X-case fi 11z has been obtained, the experimental ONP CUN~S are found to be considerably broader and the actual shape of the field dependence remained essentially unexplained for the downfield IX-case. A quick comparison of fig. 2 of [ 101 and fig. 2 of this paper indicates. however, that a quantitative interpretation of a11ONPcurves is approached, if the influence ofselective eleCtronic relaxation is included. In this section we want to give a first account of the quantitative analysiswithin the limits of the model used. While the theoretical curves of Eg. 7, havebeen caiculated for the usual case D > 0: E < 0, the samesign

is found for theD* and E* value of the triplet state complexes [9] given the usual choice of axes with the long in plane axis of the molecule termed x* Ilx and the z axis defined by the largest field separation between the two EPR (JAmI = 1) transitions. Although it is no1 necessary to fii the absolute sign, the further discussion is carried out for D*, E* > 0 which seems to be the more likely case [6]. With this assumption the theoretical results of fig. 7 can be applied to the experimental results if the following substitution of axes is carried out for all input parameters given in the figure caption:

s+y*,y-tx*,z+z+,

(13)

The two LC-cases are now H 11z* and H I/y*. The close agreement between the theoretical curves of fi’g.2 and the experimental results, in parlicular for H IIy*, is the most convincing argument for the influence of selective relaxation. Given this interpretation, the ONP results determine uniquely the type of selectivity which has to be I&=, > +Z-’ > rVX*,,*,

(14)

for all triplet state complexes observed in fluorene crystals so far. Recalling the criteria developed in section 4.2 for the cases of destructive and constructive relaxation, the former case applies clearly to the result for H II;?*, while the properties of constructive relaxation are only found in the curve for H lly*. Thus the inequality (14) is corroborated by each of the three field dependencies. A further numericalanalysis of the data shows that the inequality (I 4) implies at least one order of magnitude. As a consequence of this interpretation we are in the position to develop the best criteria for the determination of spectroscopic parameters, like the zerofield splitting and the hyperfine tensor. The zero-field sp!liting is otttained by extracting the two LC fields Hi, and Hfc from the experimental curves. In a first approximation the LC fields can be identified with the field positions of the turning point of the slope between the positive and negative contributions to ONP 3s a function of the magnetic field [6, lo]. Whilethis criterion works well for the caseH II.z*&itapplies to the case H lIy* as an upper limit of H&_ In contrast, the field position of the narrow ONPminimumin the curveH ily* is a relatively well defined feature and it is given by one of the LAC centers defined in eq. (21)

(15) This field position provides the most accurate experimental feature to derive the LC-field Hg. A further analysis requires the determination of the sign in (1.5) and hence conclusions about the hy-

perfine tensor. The numericalresults presented so far should not be applied directly because the assumption has been made that the principal axes system of the hypertine tensor coincides with that of the zero-field splitting tensor: x*,Y*, z*. This is not true for the triplet state complexes. From crystal symmetry only the x* axis is bound to be a principal axes of both tensors and hence, we have to allow the following general structure k X*X* (

0 0

0 AYP‘ AS, ZY

0 dY*z*

Az*z*

)

(16)

The influence of the off-diagonal elements in (16) will be treated in a forthcoming paper. Here we restrict ourselves to conclusions which are independent of this complicating feature. 5. I. Order ofmagriitltdc of the Il_vperfinetemor elemenrs

Modelcalculationsshow that the tensor elements have to be at least of the order of 10 G. In particular, agreement with the experimental results can only be achievedif, e.g.,A,*,,Jg;Ois larger than IO G. 5.2. Anisotropy of the llyperfne tensor The result for H IIz* can only be accounted for if the hyperfine tensor has a considerable anisotropic contribution. More specifically, the sum combination Axax* + A,,*,,= and the difference combination Ax*xz - AYp* have to differ by about a factor of 2. 5.3. 5pe ofelectron mrclearspin system

Experimentswith deuterated guest and partly deuterated host molecules [IO] permit the conclusion that the hyperfine tensor is mainly due to one of the fluorene CH, protons within the guest-host complex. A predominant hyperfme coupling with just one prom ton spin I= l/2 is also indicated in the EPR results [9]

D. Stehfik, 3.p.

C0lpo/0NP asa consequence of rhe rron~crorsi~~g m/e. If

which show a well resolved doublet for H I\x* that can be assigned to a hyperfine coupling constant A,&fl= 1 I + 2 C. Consequently, the electronnuclear spin system of the triplet state complexes in doped fluorene crystals can be well approximated by the most simple spin system of a triplet electron spin S = 1 interacting with only one nuclear spin I= l/Z. Hence, by accident the system provides an ideally simple spin system to study the feature of ONP by LAC in further detail. Furthermore, a detailed understanding of the mechanism will result in a complete determination of the hyperfine tensor, which in turn will be essential for the characterization of the unusual complexes found in doped fluorene crystals.

Appendix: Some remarks concerning the evaluation of the hyperfine tensor from ONP-datn

The theoretical treatment of ONP by LAC has shown that the relative field positions of the broad and narrow ONP contribution (:IS given by inequalities like(lS)and(76)of [l]) contain the information on the relative signs of the hyperfine tensor elements. At present, a corresponding analysis of the experimental results provides some problems which we would like to summarize briefly. The ONP-results for H II2” clearly show the broad ONP-contribution upfield from the narrow one, i.e.,

The corresponding inequality cannot be obtained unambigously from the results for H I\y*. However, the zero-field splittingI)*? E* is known from EPR [9] and provides a level-crossing field I’$ = 54 C for the system acridinc doped in fluorene; thus the ONP-results would require

However, we have shown in paper [l] that within the dmpljfying assumptions of the model the inequalities (I 7) and (18) cannot be fulfiifed simultaneously.

199

Hence, we have to ask which assumptions should be questioned in order to reach agreement with the experimental results. Most likely the following points have to be considered: (a) The hyperfine as well as the zero-field splitting tensor are assumed to have the same principal axes system. (b) Complicating features as, e.g., the influence of other nuclear spins have been neglected. (c) The small parameter E* of the zero-field splitting might be smaller than determined by EPR. Since the present theoretical analysis is limited by the assumptions (a) and (b)! the easiest why to invert inequality (18) was the use of a smaller E*-value in the calculations, in order to fit the experimental results with a diagonal hyperfine tensor due to a single nuclear spin I = 112. Thus, the parameters used for the numerical results of fig. 1 are just one alternative set to reproduce the experimental results. The influence of the other complicating properties will be dealt with in a subsequent paper.

References 111 I.P. Colpa and D. Stchlik, Chem. Phys.21 (1977) 273. [2] J.P. Colpa,KN. Hausscr and D. Stehlik.2. Naturfschg.

?6a (1971) 173!,. [3] J.P. Colpa and D. Stehlik, 2. Naturfschg. 27a (1972) 1695. (41 D. Stehlik, A. Doehring, J.P. Colpa, E. Callaghan2nd S. Kesmarky, Chcm. Phys. 7 (1975) 163. [S] K.H. Kausser and H.C. Wolf, Adv. hlago. Kcs. 6 (1976) 85. [6] (a) D. Stehlik, Habilitations Thesis, Keideibery.Uniwsi~y(19751, also published PS: (b) D. Stehlik, The Xtcchanismof Optinl Nucl~r Polarization, in: Excited States, Vol. 3, ed- E-C. Lim (Acndemic Press, !977). 17) D.A.Antheunis, B.J. Botter, J. Schmidt, P.J.F. VerbeG and J.K. van der Waals,Chem. Phys. Lett. 36 (1975) 225. [S] L.K. Hall and h1.A. El-Sayed, Chcm. Phys. 8 (1975) 271. [91 R. Purrer, J. Gromer, A. Lather, M. Schu’oer?rand H.C. WoIf, Chem. Phys. 9 (1975) 445. [lo] D. SteW& P. Rtisch. P. lau, H. Zimmcrmann and K.11.Hawser, Chem. Phys. 21 (1977) 301.