Optical nuclear polarization in molecular crystals through an optical excitation cycle

Chemical physics 7 (1975) 165-186 B North-Holland publishing Company

OPTICALNUCLEARPOLARIZATIONIN MOLECULARCRYSTALSTHROUGHAN OPTICAL EXCITATIONCYCLE D. STEHLM and A. DOEHRING Max-PIanck&utitute, Department o/Molenrbr Physics, 6900 Heidelbeg. FRC

and J.P. COLPA, E. CALLACHAN Depptment

o/Chemistry,

and S. KlZMARKY

Queen’s University,

Kingston, Ontario. ihad~

Received 3 September 1974

The theory of optical nuclear polarization (ONI’) as outlined in two earlier papers is extended andgeneralized to a de+ cription of the complete optical excitation cycle. The effects of selective population and depopulation of all electronic and nuclear spin substates of the excited triplet state are considered as well as the intluence of relvration and cross relaxation processes. Two models are discussed in detail. or.e spin system composed to two electrons (total spin S = I) and one proton (I = 112). the other of two electrons and two protons. It is argued that for most cases of interest, i.e., sufficiently large magnetic Gelds (> IOC), a superposition of al! two electron one proton systems gives a rather accurate result for the polarization in a many proton molec&r entity. The model has been applied to the ONP results observed in pure phenazine single crystals. me numerical n%ukSreproduce the experimentally observed field and orientation dependence rether closely. The only open parameter needed is the ratio of the electronic relaxation rate and the overall triplet state decay rate.

1. Introduction

the nuclear spin participation in the selective pop& tion and decay processesof the electronic triplet spin

As a result of light absorption and the subsequent population of triplet ctates, nuclear spin polarization has been observed in a number of molecular crystals El-31. The phenomenon was termed in short Optical Nuclear Polarization (ONP). It originates from an electron spin polatization or rather alignment due to symmetry determined selection rules for the singlettriplet mixing by spin-orbit coupling. Consequently, the population of the electronic spin levels in an excited triplet state can be far from the Boltzmann equilibrium and the nuclear spins are expected to participate in this polarization process via the hyperfine CoIlplhlg. An earlier explanation [2a] starts from a nonthermal equilibriumpolarization of the electron spins which is transferred to the nuclear spins via the hyperfine induced relaxation in analogy to the well known Overhausereffect. This mechanism,however, neglects

states. Although the latter are almost completely due to the electronic spin-orbit coupling, i.e., of electronic origin, the nuclear spin states “feel” a little bit of the electronic selection rules because the static part of the hyperfme coupling causesa mixing of electronic and nuclear spin states [4]. This mechanism is also able to explain the observed optical nuclear spin alignmentin zero field [5], but only if relaxation processes are also taken into account. This paper summarizesthe concept of the nuclear spin polarization through an optical pumping cycle in molecular crystals and describesthe numerical procedures to calculate the ONP for arbitrary fields and orientations with respect to the crystalline axes. The optical excitation cycle will be descnied in detail includingall electronic and nuclear spin substates and their kinetic parameters. Furthermore, section 2 defines the nuclear polarization as a result

D. Stehlik et al./ONPin molecularerystab

166

of the excitation cycle. Section 3 gives the numerical procedure to calculate the spin functions of each molecular state and their population and decay rates except for the relaxation rates which are treated in a separate chapter. Finally, model calculationsare presented and compared with recent experimental rem sulk on phenazine singlecrystals.

2. ONP as a consequence of an optical excitation cycle The nuclear spin polarization of a molecular crystal is determined by the distribution of the populations nx over the spin-sublevelsXof the molecular ground state. The ground state has the electronic spin S = 0. Therefore its spin sublevelsare of nuclear spin origin. The population nk is governed by the rate equation

P = fb,

- nAd.

If the initial nuclear polarization is chosen to be zero, the rate eq. (I) givesa simple expression for the initial nuclear polarization rate

z--I ‘6(O)aft& r=O

p=-

(I)

which is restricted to linear iens. PA denotes the pump rate out of the ground state sublevelh. N = mini is the total population in thf excited states. The

return rate or production rate R, of the ground state

sublevelXis then defmed as R,=N-‘Ck’,Ni,

i

where k!i is the decay rate from the subleveli of an excited state into the ground subleve!X. From eq.(l) it is clear that by an excitation cycle the populations nk are removed from a thermal equilibruim only if the pump rates and (or) the return rates are not equal for all sublevelsX,i.e. ah nuclear spin states. Equal rates PA and R, are, however, given for the excitation of electronic singlet states and hence their excitation cannot produce any nuclear polarization. The situation is different for an interim population of excited triplet states during an optical excitation cycle. It is demonstrated later that the return rates R, depend on the sublevelAin this case.

In most ONPexperiments so far, the excitation cycle starts with light absorption in the excited singlet states, i.e., equal Ph. The change in the population nXis then determined by the R, from excited triplet states only. The nuclear polarization is generally defined by the population differences nx - nAr. The proper summation and normalization of these terms may be written in a generalformas

R,t) .

More explicit expressions(3) and (4) can be given for particular nuclear spin systems. In the simplestcase it is assumedthat the molecule contains only one nuclear spin I = l/2. In a magnetic field the nuclear polarization is defmed as the difference between the populations on the nuclear Zeeman levelsX= +, which are characterized by the quantum number MI = c l/2: k-n T

dnJdr = -P,-,n, + R,N ,

(3)

n++n_

-

.

04

the initial nuclear polarization rate is then related to the return rates by

With the proper normalization

fi(O)=R+-R_.

@a)

The eq. (4a) provides then the connection between an experimentally measurablequantity i(O) and the return rates R, in (2) which have to be calculated from a model of the optical excitation cycle. 2.1. Tire model of the optical ercitah’on cycle

If higher molecular states are excited by light absorption, the predominant population is carried by the lowest excited singlet and triplet state due to the Qst radiationlessand vibrational decay processes. Hence it is adequate to restrict the energy diagram of a molecule to these excited states. Together with the transitions among these states, it constitutes the so called Jablonski scheme (fig. I) describingthe luminiscenceproperties of excited molecules.The energy schemehas been extended to include also nuclear spin sublevels.For simplicityonly one nuclear spin I = l/2 is considered. The nuclear spin energy splitting is largely exaggeratedto be made visiblein the drawing.A molecular singletstate ha; two sublevels indexed X,while the triplet state has i sublevels,in total (2s + 1)(2J t 1) = 6. Since the Z-electron and l-proton system constitutes a Kriunerssystem, the twofold degeneracyof the levelsis only lifted in the pres-

D. Stehlik et aL/ONPin molecular crystuls

Fig 1. Jablomki s&me including nucleas substates. of a magnetic field. The zero-field case has been described before [S] . It has been well establishedthat in molecular crystals the spin forbidden transition between electronic singlet and triplet states, often called intersystem crossing, is partly allowed due to the singlet-triplet mixingby the +m-orbit coupling*. The intersystem crossing can be highly spin selective due to symmetry ence

selection rules for the spin orbit matrix elements. As

a consequence, the electronic spin sublevelsin the triplet state are populated and depopulated with different rates. The result is the well-known optical electron polarization (OEP) which is observablewhen the electronic spin relaxation is sufficiently slow. Although OEP does not depend on the nuclear spin explicitly as stated above, an implicit dependence is introduced if the effect of the hyp?rfine coupling is included. Due to the hyperfrne coupling the triplet substates i are linear combinations of the pure electronic and nuclear spin product states. Combined with the selective intersystem crossingrates, generally different population rates (Pi>and decay rates (ki) are predicted [4]. The population of the substates i is then governed by the following rate equations dNi/dt = Pi 1 kjVi

167

The Boltzmann distribution N,! -IV; might be neglected if population differences far from the thermal equilibrium are considered. In the above formula the quantities are normalized to one created triplet state by setting the sum of the population rates equal t0 Unity: Ei Pi = 1. NO higher order decay processesare included in (5) assuming sufficiently low triplet concentrations to neglect them. With this model the optid nuclear polarization can be calculated by the followingstepwiseprocedure: (1) Determine the rate constants in (5). (2) Solve (5) for the equilibriumpopulation Ni: (3) Insert Ni and the decay rate components k: into (4) to calculate the production rates and hence the initial nuclear polarization rate.

3. Numericalprocedure

3.1. Spinhantiltonian and dingoneliration In order to determine the population and the transition probabilities of the molecular triplet state sub lebels, the spin hamiltonian has to be specified. Q,, The

=9$j +sl; +.aa$s +sr; +slg.

(6)

followingcontributions are considered:

slD P s, - De-S2 = D(S; - $S*) + &

- Sj)

(7)

is +Aedipole interaction between the two unpaired electron spins S, and S2 which make up the triplet state. S = St t S, is the total electron spin. The principal axis system of the dipolar tensor 0, istermed x,y, z and coincides usually with the molecular axis system because of symmetry requirements.

is the electronic Zeeman energy diagonalizedin its second form. In agreement with the experimental observation, it is legitimate to assumethat (7) and (8) have the same principal axis system, i.e., u = X,y. L.

wi/ denote the relaxation rates among the sublevels. l

The wrre5ponding matrix elements fall in tie range

2 0.1 cm-’ in the case of aromatichydrocarbon and are largerby at least a iactor of 10’ to lo3 than otha a~ntrtbutions. a.&, due lo the hypetic coupling.

is the hyperfine coupling among the electron spin S = 1 and the total nuclear spin I. A singletensor A can be used only in the case of equivalent nuclei.

D. Stehlik et aL/ONP in moleculurcryrtals

168

Furthermare.the assumptionwillbe madelaterthat the hyperfme interaction is dominated by ils intra-

Table 1 Nuclearspin functions used

molecular part, i.e., only nuclear spins of the molecule carrying the triplet excitation are taken into account. RZ* = - gJ#i*

I= - g,@f,+

00)

is the nuclear Zeeman coupling where a scalarg-factor can be assumed.It is customary to definea frameof reference with 2’ IIH, the quantization axis with respect to which the final nuclear polarization is defined. The complete magnetic field oriented frame of reference is called x’J’, z’. Finally, a zero field nuclear spin energy must be considered. It is dominated by the nuclear dipole energy in the case Ofii = l/2 nuclear spins.

~i;=CIi.Dn.Il. r+j

(11)

In the case of two equivalent nuclear spins Ii = l/2 (f = 1) eq. (11) can be transformed to a diagonal form sl”o = ,‘I@2- 315) ,

One spin i= l/2

%’

Q’

and B’

TWOS@IBfi=1/2(f=l~df=O)

~‘a’, $$, -jj (a’$ + .P$) = Tzl and S” =$j (a’$ - B’a’)

aandS

spin functions introduced here is collected once more in table 1. The total wavefunction has to include also the orbital part cpand can be written then in the following form for aUsubstates of fig. 1. Electronic singlet states: A= (U + 1) sublevels (13)

Jl& = Lps,Se% 3

(12)

with the constant b =&$G and the principal axis z* lying along the internuclear distance vector f. In order to diagonalbe the total spin hamiltonian (6) a convenient set of basis functions has to be chosen. With respect to the electronic spin energy part most often the eigenfunction of the zero field energy (7) are used and written here in the notation c@ = x, y, 2). The correspondingsinglet state spin function is termed Se. \ The choice pf the nuclear spin functions should be kept variabledependingon the total nuclear spin system with two huclear spinsI = l/2 it is convenient to use the anal0 us set of spin functions in both the electron and n clear spin system 151,i.e., uV= T,” (v= x,y, 2) an T S”. In view of the observation of the nuclear polarization in a magnetic field, it seemspreferable to use another set of nuclear spin functions based on the &man eigenfunctionswith the magnetic quantum numbers mi= ti and -1 for a I = l/2 spin. In a common notation, the corresponding spin functions are termed a and P. Lateron the molecular frame of axes x,y, I as well as the magneticallyoriented frame x’,y’, z’ are used and the correspondingspin functions are distinguishedby the prime. The set of

~~,=~~,Seo~=IPS,Se

c 9;o, Y

04)

Electronic triplet state: i = (2s + 1) (2f + 1) sublevels

(!5) The important mixing coefficients CL can be obtained from a diagonalizationof the total spin hamiltonian. For the zero field case with no Zeeman terms this has been carried out earlier [5]. Here the presence of a magnetic field is assumedsuftkientiy large to neglect the dipolar nuclear spin energy (11) in the spin hamiltonian. The numericalprocedure is restricted to the simplestspirrsystems, i.e., the triplet electron spin S = 1 is allowed to interact with one or two equivalent nuclear spinsI = l/2. Ibis restriction is necessary because otherwise the theoretical results depend on to many open parameters which would limit the value of the model. With respect to the product states q u, the matrix form of the spin hamiltonian (6) is given in tables 2 and 3 for the two casesof table 1. Two additional assumptionshave been made:

D. Srehlik et al./ONP

in molecularcrystals

169

Table 2

Spinhamiltonian for the two-ekctron. one-protoncase 1 1

D-E-

&?,tff

2

2

S

4

5

6

- i($l,,+~,,Bgff)

isyy BeqH

-fg&,(P-WH

0

f A,

D+E-&I#

-ig&~H

0

-~g,B,(P-iqw

-h ‘iA xx

3

-fg&H

4

-SAyy

Sk

-&&(P-igw

D- E+;g&,rH

N~Azz-gi&H~

kyyScqH

D+E+&,&tH

- igxwUePH

5 6

&,&rH

‘Ihe lower triangdar

part

of the matrix can be found by the relation Hl= Hii

(1) The principal axis system of the hyperfiie tensor (9) is identical to that of the electronic dipolar (7) and g tensor (8)t. (2) The nuclear spin function Sn has been omitted in table 3 because its energy contribution is unaffected by all magnetic interactions in (6). The presence of a nuclear singlet state in the excited triplet state could, however, alter the population on the other spin states via relaxation and hence affect the fina nuclear polarizarion. This effect is probably very small and hence it has been neglected in the model. The magnetic field components with respect to the molecular axes x,y, z are expressedby

H,=pH,

!$=qH,

H,=rH.

p, q, r are the directional cosinesof the field axis with respect to the molecular axes. H is the field amplitude.

In summary, the followingparameters have to be inserted into the matrix: electron dipole interaction: hyperfme interaction: electron Zeeman energy: nuclear Zeeman energy: field value and its orientation:

D, E J%VJ$ A,, g,, gY,,,gzztP

(16)

Gifln

H, p, q, r.

The diagonalizationof the matrix yiel_dsthe eigenvaluesEi and the mixing coefficient CL as defmed in

(15). t In the general case the two principle axesof (9) within the molecular ptanearerotatedwithrespectto the molecuk axissystemand the c~~nespoodingtramformation is car&l out easily.

3.2. 7?rerate constants The population rate Pi of each subleveli in fig. 1 is calculated as in ref. [4] : (17) ‘&e pP (l&p,, = 1) denote the relative population rates of the electronic spin level c and are related to the spin-orbit matrix elements for the singlet-triplet intersystem crossing.In many molecules they have been measured directly. P: is then a populationrate component whichgivesthe population rate of the subleveli if only one c-level is allowedin spin-orkit mixing such that for the particular b’ = 1. TheP: .Ffi : are given by the mixing coefficients c,,,[4]

(18) It is assumedthat all nuclear spin levelsin the excited singlet state are equaily populated; this situation has been chosen to be the initial condition of the ONP experimt&. Then the generalproportionality constant C, is givenas the relativepopulation in each of the sublevelsA,i.e., equal to the inverse number of sublevelstaken into account. Cl = l/2 in the case ofonenuclearspinf=1/2;C~=II3incaseoftwo spins1, = l/2, neglectingthe nuclear singlet spin fimction. The orlhonormality of the mixing coeficients z&J2 = I provides then the normalization XiPi = I and all results derived from here are normalizedto one createdtripletstate.

170

D. Srehllk et aL/ONP in molecub crystals

The decay rates &in the rate equations (5) are obtained similarly. In addition, each component of ki is needed in ki=

Ck’ A

I’

(19)

e.g., k; = k’, + 6 in the one spin I = l/2 case. Analogousto (17) one definea

The k,, are the absolute decay rates of the electronic spin twels q while k’, (!Z&,= 1) are the relative ones. k = Z,,k, is introduced as a total decay rate which is related to the phosphorescence lifetime by rph = 31k. The decay rates k,, as well as the relative population rates p,, have been determined experimentally for many molecular triplet states (e.g., table 5). In most experiments the nuclear polarization is generated on the nuclear Zeeman levels of the molecular ground state, hence these states are givenby the magnetic field oriented spin functions uyl of table 1, i.e., X= v’. The determination of the normalized rate components k’,, in (20) requires a transformation from the molecular frame of axes x, y, 2 to the magnetic field oriented one x’, y’, z’ which is carried out in the appendix. Only the result is givenhere. Case 1: onespinf = 112

Case2: two equivalent spinsI, = l/2

GM

D, Stehlik

With the orthonormality

et al.lONP in molecularcrystals

of the mixing coefficients

CL it is verified easily

that the rate components Qti = k’,~ are normalized,i.e.,

171

rates (2). Furthermore, the measurementof the initial polarization rate keeps the number of multiple excitation cycles negligiblefor any individualmolecule and the assumptionof equally populated nuclear substates in the excited singletstates (see derivation of (18)) is justified. In summary, for any parameter set (16) the numerical procedure described here provides the foIlowing production rates

y/=1. It is convenient to use only relative population and decay rates in the rate eqs.(5). With the total decay constant k = Z,k, defied in (20) it can be achieved by substituting in (5) ki + k; = C GPih 9 Ni j Wvi1 Wii~ rvij/k . (23) P

In this form it is also demonstrated that the tesult dependson the relaxation rates only with respect to the total decay rate. The last parameters to be determined in the rate eqs.(5) are the relaxation rates wii_Thy depend on the kind of relaxation mechanismand are given for some relaxation models in the next section. Introducing all parameters into the rate equation, the next step in the numericalprocedure is their solution. In a fm approach the equilibriumsolutions Nl{e) from dNr/dt = 0 are used based on the following ponsiderations.In an actual molecular crystal the dncentration of the excited triplet states is rather low. On the other side, the nuclear spinsof all molecules constitute a large thermal reservoirthe temperature of which is alwaysin an internal equilibrium,in other words, the equilibriumtime constant, i.e., the spin-spin relaxation time Ta, is very short compared to all time constants changinglocalIythe population distribution on the nuclear sublevels,here in particular by an optical excitation cycle. Therefore, it is not possible to separate out a nuclear reservoir made up of just those nuclei which belong to the molecules involved in an optical excitation cycle, rather the only measurable quantity is the polarization of the total nuclear spin reservoir. Furthermore, it is experimentally observed that the optically generated polarization growswith a time constant still very slow compared to all time constants in(S), and to a very good appoximation the equilibriumsolutions of (5) can be used to calculate the nuclear polarization rate (4), i.e., the production

and the initial polarization rate follows from an appropriate summation of these rates as given in (4). It should be noted that so far it is assumed that the optica excitation cycle is completed at one molecule. In actual experimental situations it is possible, however, that population and decay of the triplet state take place at different molecular sites. Energy transport via guest-host transfer or due to exciton character of the host triplet state are known reasons. In this case different parameter sets (16) must be allowed to calculate the mixingcoefficient &, required in the expressiorr for the popuIation and decay rates.

4. Concepts of Axation 4.1. Relaxation induced by pun: elecfronic rpin relaxation and the level mixing by hyperfine interaction

The mixing of the pure product states cuV by the hyperfme coupling induces relaxation rates Wij among all levels of fig. 1 if a relaxation mechanismof purely electronic origin exists among the electronic spin states. As a convenient starting point, the relaxation rates w$,,*among the zero field spin functions c are chosen which may be written as

‘ri, is the spin part of the time dependent relaxation operator. The time dependence results in a spectral density function J(U) which has to be taken at the transition frequency f+*. If a set of zero field relaxation rates wL* are known, the general relaxation wd due to thii relaxation mechanism can be expressedas a function of the wh+ by writing

D. Stehlik et allONP

172

wi/= I($jlli,l

$$I2 J(Wii>*

(25)

The r$dxation rate operator acts only on the spin part ~W‘JWT+7,of the wavefunction $,. inserting the mixing coefficients cf known from the diagonabzation procedure one obtains for (25) w~=w!!Ytw$v$z,

(26)

with

The frequency factor takes care of the frequency dependenceof the spectral density function J(W). In summary, eqs.(26) provide expressionsfor the relaxation rates WVat arbitrary field orientations assumingan electronic relaxation mechanismwhich is transferred to all hyperfine levelsby a static hyperfine interaction. Only three new parametersare necessary, the zero field relaxation rates w;. or their relative value (23) w# with respect to the total decay constant k. The wewlaare not just freely variable parameters,rather experimentalevidenceis available concerningtheir origin. Later on, particular emphasis is givento the electronic relaxation mechanismas a consequenceof a delacalizationof the triplet excitation, i.e., an exciton motion. Therefore, a short review is givenconcerningthe zero field relaxation rates due to exciton motion. F(Ujj)

=J(W;j)/J(Uw~)

4.2. Electron spin relaxation by tr@let exciton motion The electronic spin lattice relaxation rates of excitattic.triplet states have bsen measuredat room temperature only in a few molecularcrystals, e.g., anthracene, naphthalene [6j, and dibromonaphthalene [7]. The resultsare well explainedby the assumptionof incoherent excitons [6,8], i.e., the excQon motion can be describedby the so-called*‘hoppingmodel”. The hopping introduces an electronic relaxation mechanismby modulatingthe magneticinteractions, in particular the zero field splitting,if the crystal contams inequivalentmolecularsites. In most casestwo molecularsites A and B are found with the followingdifference term among their respective zero field energy5Xb(eq. (6)):

in molecular crystals

ASXD=%;(A) -@c,(B) .

(27)

A(jr, = k, can be identified with tlte relaxation operator in (24) becausethe exciton hopping modulates this interaction component as it is turned off and on statisticallyat a givenmolecular site. It is not necessaryto go further into the relaxation theory, rather is it possibleto developgeneralrelations among the wh* (24) by inspectingthe form of ASr, (27) in actual molecularcrystals. Some systems as anthracene, naphthalene,and fluorene have the property that the long molecularin-planeaxis, often termed x, has the same orientation with respect to the crystalline axes for aUmolecular sites. Consequently, the L-componentof the zero field energy (9&), I@ mainsunaffected by the exciton motion. Hence the relaxation operator can be written k&=A9$,aSz.

(28)

The specific relaxation question here is as follows. The popufation on the three electronic zero field levelsc is considered in one type of molecular sites. Whatsort of change in population is expected if the triplet excitation is generated at the molecular site directly or via excitation from all other molecular sites through the hopping? The difference can be described by additional_relaxationrates wL* in the latter cass introduced by R, (28). However,(28) does not introduce relaxation transitions from the i”$evel sincef&c = 0. Hence only an additional relaxation rate w& is predicted due to the exciton hopping. In other words, an exciton motion, which is connected with a modulation of the fine structure, generally inducesa selectiverelaxation among the electronic zero field levels,such that one relaxation transition wh* is preferred as compared to the others. Iv;. > wew.*) w;rp

.

(29)

4.3. Relaxation by time dependent hyperfine intemcti0n As in the case of any fluctuatingmagneticinteraction, a time dependent hyperfine interaction induces relaxation transitionsamong the spin sublevels.In particular, the exciton motion describedabove con-

D. Stehlik et al.IOhT

stitutes a time dependence of the hyperfine coupling. As a consequence, relaxation transitions in the combined electron and nuclear spin reservoirare possible. Usuallythe exciton motion is fast with respect to the relaxation rates and hence relaxation must be considered between spin levelsaveragedby the exciton motion. Since the exciton motion is also fast as compared to the reciprocalhyperfme splitting, the static hyperfme interaction is averagedto zero, and the total spin function can be givenas a product state of electron and spin part. Instead of the spin function in (15) (30) it is appropriate to wtite

in molecularcrystals

173

The last relaxation rate is purely electronic and usually dominated by other relaxation mechanisms than the hyperfme induced one. The r&xation rates wti due tow; transitions can be treated as in section 4.1 and can be omitted from the further oansiderations. The task is now to find expressionsfor the relaxation rate wi/ among the levelsof fig. 1 as a function of the relaxation rates wo, wl, w2 (32) which are known experimentally in some cases. The spin operators in (32) are defined with respect to the magnetic field oriented frame of axes. Hence the spin functions (30) should be transformed to a basisof magnetic field oriented spin functions. The transformation for the nuclear spin part is easy with the transformation given in the appendix

(30 Relaxation among the product states (31) has been describedin ref. [ZaJ. Since the diagonalizationprocedure necessary in this paper results in the.spin functions (30), they are used here as an approximation for the correct functions (31). The large uncertainty in the relaxation model justifies such a purely practical approximation. Nuclear relaxation in molecular crystals due to time dependent hyperfme couplinghas been studied in some systems [ 1,a] . ‘Ihe frequency dependence of the nuclear relaxation among Zeemanlevels w,(w) a J,(W) is proportional to a spectral density determined by the properties of the statisticalexciton motion. A rather general diffusion equation for the motion has been included within this relaxation model [IO]. The model givesthe nuclear relaxation rate w,(w) as the sum of relaxation rates among the electronic and nuclear Zeemanlevelswhich might be written as Im$ and Im$. The rate components and the respectivespin operators inducingthem are: w(#,I*):

Then eq. (30) changesto (33) with

The transformation of the electronic spin part can be treated differently via the spin operators. Analogous to eqs. (25) and (26) one can write for the relaxation rate induced by hype&me coupling Wij= lC$ilSX 121$j)12 tI(J,Is

I

WO

i1~~1*w~+l~~~Is’II~~1~w 1ztj Lt.t j

1.

Inserting(33) one obtains

(Am,=+-l,Am~=rl),

w#* I*): (bm~ = AmI= + I), (32) w,(S&*):

(Ams = 0, Lvn, = * 13)

Wi(S,lze):

(Pms=+l,hmi=O).

S, = S,a f $1;

frame of axes.

x’,)l’,

t’

is the magnetic field oriented

The matrix elements in (34) can be calculated by expressingthe spin operator through a coordinate transformation on the molecular frame of axes (35)

D. Stehlik et aL/QNP in mole&v mstak

174

p’ = x’,y’,z’,and p = x,y, z. The transformation matrix contains the directional cosines amongthe different axis systems and is given in the appendix. The electronic zero field spin states G are e&en. functions of the 4 component with

where

4Ni N~e)*(2Y+

1)(2l+

I)’

As a consequence, the return rates (2a) reduce to

0% where K, X,p = x,y, L and cyclic. Hence

= I-i(c$ X cja~).12,

(36)

i.e., the matrix element is given as the kcomponent of the vector product of vectors like

The mstrix elements in (34) are then given by

and become equal for each ground state level due to normalization conditions. As stated before, no nuclear polarization can be generated in this case, even if the relative electronic population rates + (17) and/or decay rates k,, (2 I) are highly selectrve. Even a selective zero field relaxation (3a) equalizes the populations according to the mechanismof section 4.1 provided the dominating relaxation rate is sufficiently fast and hyperfine coupling strong enough to assurewii b ki. According to eqs. (4) and (4a) the busing-up time constant of the optical nuclear polarization is related to the difference in return rites R, - R,. . Hence another consequence of fast relaxation within the optical ~rnp~g cycle is a slowdownof the build-up time of the nuclear pola~tion.

4.5. ctoss relaxation

withu,u=x,y,z.

The relaxation rates wi, induced by a time depen dent hyperfme ccupling are then expressed by the high field rates wo, wl, y2 and numerical factors dependent on the mixiig coefficients CL known from the diagonalizationprocedure and the directional cosines involvedin it. 4.4. Efpct

offat tehution in the optical excitation

cycle Fast relaxation rates wd * ki compared to the decay rates (19) equalizesthe equilibriumpopulation on ah triplet state sublevels

The term cross relaxation is used here for the energy* transfer within the spin system due to nearly equal& spaced pairs of levels.The correspondingmutual spin transitions conserve energy and hence can have a ratber high probability. Typical cross relaxation processes are represented in fig. 2. Only four levels of two molecular level schemes(printed and unprimed) are shown. Simultaneoustransitions take place in the primed and unpruned level schemes.The condition of energy conservation is fulfdied within the overlap of the limeshapesof the two t~tio~ at the frequency n1 and ~2. The overlap amplitude is expected to be proportional to the cross relaxation rate and is itself a lineshape function f(u, - ~3 depending on the difference of the transition frequencies. As a consequence of the cross relaxation, the pop~ations Ni on the triplet state sublevelsin fig. 1 can change. The CIOSS relaxation terms to be added in the rate eqs. (5) have the followinggeneral form [ 1l]

D. Stehlik et aL/ONP

-

in molecular oystafs

-1’

,’

I’

E jv2 :

12(vJ ml

iol

(b)

Fig. 2. Two representativeexamples of cross relaxation Case a: Mutualtransitionsoccur in two spin systems with the same energy level scheme.The transition frequenciesY, and v1 between the degeneratelevels m = n ahd m’ = n’ and the non-degeneratelevels are assumedto overlap within their respectivelinewidths. Caseb: The two spin systems have generallydifferent energy level schemes. Here the transition frequenciesY, and u, between different pahs of levels 0verIapwithin their respectivelinewidths.

175

Eq. (41) ad& another Lear relaxation term to the rate eq. (5) if 2, N, is not changed by the cross relaxation process. This assumption can be fulfilled if only pairs of levels (i, j) are close enough in energy such that ouly the lineshapef(vf - U& =i(~i=)has a sizeablevalue while the values of&J and&J can ne neglected for all n. N, in (41) can be approximated by the pop?llation distribution without cross relaxation provided that the decay constants ki and ki are not very different. These serious restrictions limit the approximation (41) very much, but since the cross r&&ion term is reduced to a linear term, the computational effort is minimized. The cross relaxation in its general form (40) adds bilinear terms to the rate equations. Therefore, they can be solved only approximately. Most useful are iterative procedures starting from the solution of the linear part of the rate equations. 4.6. 77lecross relaxation Iineshqe factor flvl=flq - $1

- (Ni Nml- NnN,*)adI -

(40)

The probability functionf(v) =f(vl - u2) which The negativesign of (40) denotes the cross relaxaenters the cross relaxation term is given by the protion process with simultaneoustransitions from level bability of a mutual spin flip in the two level schemeswith hv being the energy that is left over. i to R and level m’ to I’ (solid arrows in fig. 2), while the positive sign characterizes the inverse set of tranThe maximumvalue off(u) is hence reached for sitions (broken arrows in fig. 2). v = 0. Usually,f(v) is approximated by the convoluWr-uis a proportionality factor determining the tion of the liieshape function of the transitions 1 and 2 strength of the cross relaxation. Due to cross relaxation the spin system relaxes towards an equilibrium populaf(v) = jf,(Y?f*(v - 4dy’ (42) tion which might be different from the thermal equilibrium because it has to be consistent with the f(v) can be givenexplicitly if the cross relaxing transiadiabatic character of the cross relaxation process [ 1l] tions have lorentzian or gaussianlineshape, e.g., For large population differences the equilibrium term in (40) might be neglected as well as the thermal equilibrium in the linear relaxation tern of (5). The cross relaxation leavesthe sum of the populations on the cross relaxing levelsunchanged. or The general form of the cross relaxation term (40) can be simplified for the cross relaxation process of uz 1 f~,#‘) = 01~~ exp - g (43) Cg.2, case a. First, the population of the two spin ( 23 1 level systems can be assumedto be the same. l?qualizing the indicesI= I’ = j and m’ = m = n’ = n (40) simplifies where 6 1 is the half width at half intensity with the after summation over all states n # i, j to relation 6 1 = 1-180, for the gaussiaulineshape. Usingthe theorem that the Fourier transform of a convoluted function is the product of the Fourier (41)

D. Stehlik

176

et al.fONPin tzvlecubrcrysralz

transforms of the individuallineshapes,it is easily verified that the convolutions of two lorentzian and two gaussianlines respectively are f;(u) = c,

-L A$’

f&9 = CG exp [- $),

with 6=S1+62,

with Oz= o: t o$ .

(44)

(451

The proportionality constants CL and C, are given by the product of the corresponding constants of the individuallineshapes. Note that by the help of the above theorem the convolution of a lorentzian and gaussianlineshape can be numerically constructed quite easily. For a given molecular spin system, as in cross relax% tion case a (fig. 2), f(u) takes a certain value depending on the difference frequency y = v1 - v2. In case b (fig. 2) the cross relaxation effect is usually observed as a function of an external parameter (magnetic field or orientation) and thus a function of the frequencies v1 and 9. The lineshape function f(v) (42) can then be observed di~ctly. It should be noted that the definition (42) of the convoluted function concerns only the lineshape or linewidth while it is independent of the chosen center frequenciesof the individual1inesfI and fz. Suppose we have a lineshape functionf,(v) centered around an arbitrarily chosen zeropoint and another Iineshape function&(v - 5) centered around a frequency b in the chosen frequency scale. The convolutionf(v) off,(u) and&(v - b) is according to (42) given by s ~I(u’)~~(u- v’ - b) du’ . This new function has v - b where (42) had only v and hence the whole function is shifted over a distance b without changingits shape however. Hence the lineshape can be obtained without bothering about the positions of the centers of the originallines. The center of the line must be determined independently Ram the averageof the centers of the lines 1 and 2. A remark seems to be appropriate concerning the fact that the cross relaxation transitions constitute bimolecular processes.Those have been exduded from the rate eqa. (5) with respect to bimolecular excitation and decay processesof the electronic triplet

state which are known to occur at rather high excitation concentrations. In contrast, the cross relaxation mechanism is expected to be effective already at much lower concentrations. It is due to the magnetic spin spin interaction as are the T2 relaxation processes. In fact, the electronic spin spin relaxation time G is a lower limit for the cross relaxation time. Since the electronic triplet spin system is known to reach internal equilibrium (spin temperature assumption), even at very low concentrations due to the T2 relaxation it seems reasonableto assumeefficient cross relaxation in the whole concentration region typical for an ONP experiment and well below the concentration region of bimolecular population and decay processes.

5. Numen-calresults 5.1. Input values Large optical nuclear polarizations of protons have been found so far in single crystals of anthracene, fluorene, and phenazine. The parameters needed in the hamiltonian (6) concerning the crystal structure and the molecular interactions are collected in table 4 ilS far aa they are experimentally known. The data of the magnetic interactions have been determined for localized triplet states, i.e., the molecule of interest is doped in a suitable host crystal. Although solvent effects mainly on the electronic parameters are well established,they will be ignored for the fim order calculations intended here. In order to set up the eqs. (S), the kinetic parameters of the optical excitation cycle must be specified. In recent years, population and decay rates have been measured in a large number of molecular triplet states mainly using the method of optical detection of the magnetic resonance transitions (ODMR) [ 121. ‘Ihe observed rates are interpreted as rate constants of the purely electronic spin functions, although the latter are mixed by the hyperfiie coupling. Since the deviation is expected [4] to reach only the experimental accuracy level of a few percent, th lxperimental rates can be identified with the population ratespr and decay rates $, introduced in section 3.2. Experimental rate constants are collected in table 5 for the molecular triptet states involved in ONP experiments.

x

2.003

2.001

SYY

bzt

-8.0

A,,/h (106W

Ref.

-3.8

(106Hz)

AY,lhUOW

Proton

- 0.0

2.005

gxx

A&

I171

Ref.

1,4,5.8 a

0.330

-E/h (1091-Iz)

podtlon

2.232

+0.5709

C’

D(IO%z~

(+I0.698S

b

2 3

paramctern

fO.4313

x

interaction

a

Directional cosines

L-

TY

Experimental

Table4

1171

I191

0.325

+0.4168

#1.2060

-0.8854

Y

Phenazine

7

6

20.5

-2.1

..~

20.5

-3.1 (1cNI

1191

20.5

-1.0 (CHI

7 CM,

2.003

2.003

2.003

1191

0.090

3.182

1

0

0

X

2,3,6,7 0

pjo.7074

-0.6852

@‘736

z

96

lf 5’ L’

-15.8

-9.9

-3.8 (CH)

-13.2 (KH)

2.2’

Wl

0.084

3.183

0

+o.S708

oo.821 1

z

1181

-26.3 (LCH)

-7.5 (CHI

IUOlllllliC

4,4’

0

+0.8211

($0.5708

Y

w5

3

4

Fluorene

-16.6

-NJ

-24.8

9,lO meso

2.003

2.003

2.003

[I71

0.253

2.147

-0.8600

-0.1274

-0.4941

X

-.

Y

.

1171

-8.3

-4.0

-12.4

1,4,5,6 a

-0.3149

-0.8944

fio.3175

-3.8

-5.7 (KH)

-1.9 (Cli)

2,3,6,7 B

I61

0.251

2.082

-0.4015

-0.4287

+o.a093

Z

-_”..--

7

8

3 10

6

2 4

Anthracene

..”

“..

,... -_

D. Stehlik et al/ONP in molecular cryrtaols

178 Table 5

(17) and decay paramcte~la (20) of the electronic triplet spin states

Rdati~epoptition

PX

0.39

PY PZ

0.55 0.06

0.33 0.33 0.33

0.33 0.33 0.33

CL% 0.01 0.03

0.90 0.05 0.05

0.50 0.25 0.25

0.90 0.05 0.05

0.90 0.05 0.05

0.60 0.20 0.20

0.40 0.36 0.24

k’, + &I

0.60 0.34 0.06

0.64 0.33 0.03

-

0.90 0.06 0.04

0.81 0.16 0.03

-

0.88 0.08 0.04

-

0.10 0.80 0.10

largest -

Rd.

1131

I141

I161

il31

1141

1161

I131

I161

1151

1161

Host Acaptor

diphenyl -

diphenyl

fluorene -

Ruorene

-

fluorene

-

The largestuncertainty in the kinetic parameters concerns the relaxation rates. Within the relaxation model (section 4.1) the relation (29) among the zero field rate constants w$ is sufficient. In fluorene and anthracene crystals the exciton hopping among inequivalentmolecular states does not affect the c-level, and the relaxation rate w& is predicted to be predominant. The phenazine crystal structure seems to favour the hopping between equivalent sites since thg are arrayed in chains. Neverthelessthe electronic relaxation rate can be dominated by the remaininginterchain hopping as demonstrated in the dibromonaphthalene crystal 171. In the case of the phenazine crystal, w& is then predicted to be the predominant relaxation rate. 5.2. ONPas

a function

of the magnetic

field

and its

orientation

Most experimental information on the optical nuclear polarization was obtained so far from the magnetic field dependence as well as its orientation dependence with respect to the crystalline axes. For crystal cutting reasonsthe orientation dependence is most conveniently measured in three perpendicular crystalline planes. Hence the numerical calculations

fluorene

diphenyl fluorene

-

fluonne

fluorene fluorene

have been performed accordingly and the magnetic field orientation has been varied stepwise in these crystalline planes. In most cases,the field orientations have been limited fi.uther to the quasi mirror planes because all molecules are then magetically

equivalent

and the same mixing coefficients c’~ can be used throughout the whole excitation cycle. 5.3. Restriction to the onespin I = 112case The molecules(tabte 4) involvedin ONP experiments contain alwaystwo or four proton spins in equivalent positions. Even if only equivalent protons are considered, it seemsnecessary to work with a total spin containing severalnuclear spins. It is, however, possible to restrict the calculationsto the S = 1 and one I = l/2 spin system if the proton spins are decoupled from each other. ln order to investigatethis point, the ONP was calculated with the two electron one proton program described in section 3. In addition, calculationswere made comparinga two proton case with a four proton case (two pairs of protons, not necessarilyequivalent) [ZO].The results ahow always that in fields above the 20 G range the overall ONP is additive in the resultsof the individual proton cases to a very good approximation. The investigationof the hyperfiie pattern of the triplet state ESR lines as a

D. Srehlik er aL fONP

function of the magneticfield comes to mind where a similarfinding is observed [21]. Even though the nuclear Zeeman interaction is smallerthan the hyperfine coupling in the intermediate field range (5 G
crystals

Amongthe systemswhich allow a considerable proton polarization at room temperature by light irradiation, phenazine is so far the host whose protons can be polarized most efficiently without the need of suitable guest moleculesin the crystalline lattice. In very pure phenazine crystals grown by sublimation, the only sizeablepolarization is found for one particular field orientation in the ac-plane. A typical polarization result as a function of the polar angle with respect to the c’-axis(lab-plane) is shown in fig.3. In addition, the figure presents the field dependence of the peak polarization. The main experimental features can be summarizedas follows: (a) The peak polarization (ff = 3.4 I&) corresponds to an equilibriumpolarization Q2a) of pL = 2.8 X 10s4

in moleculor qvsrals

PL Cl65 L

3-

j

8-

I' I I.

2-

j\ ,b I

;,

: .

I

‘

i

i *.

. .-.* .. . -.. l

1I

. -.

10

..

I 95

...

l

. l

*

1

I

HP CkGauss7

-

(

,

,,

,

,

,

,

,

123156789 @I

Fii 3. Optical Nuclear Pohrizatian at roam temperature in phenazinssingle crystal grown by sublimntion (unpubbhed resultsof P. Lpu).The initial nudeax polarization rate i(O) as

~~theeguilibriumpo~ti~np~=~O)T~~oreplatted as a function of (a) the orientationof the ma&ncticiicld ffP = 3.4 Hi in the ac-pbe (the crystal has been rotated around the crystiline b-axis). (b) the field strength HP (oriented for maximum ONP).

with a polarization rate$(O) =pL/TIL= 0.9X 10’6s-1.

180

D. Stehlik et PljONP in molecularcrystals

A preliminary measurement of the quantum efficiency givesa polarization rate per absorbed singlet state [2b] of d,(O) = 3 X 10s4 s-l. Dependingon the quantum efficiency of the intersystem crossing processbT(0) 2: 10m3s-* is expected for the polarization rate per created triplet state in eq. (2). (b) The peak polarization in fig. 3a is observed very near the orientation of the field which is most closely parallel to the shortjn plane axis, i.e., the molecular y-axis, with an angleof 25” to the a-axis according to the crystallographicdata. (c) Fig.3b showsa maximum of the polariza!ion around HP = 1.5kG decreasingslowly with increasing field. The characteristic orientation dependence of fig. 3a is nearly unaffected up to fields of 70 kc except for a broadeningof the peak. The simple orientation dependence in a plme with equivalent molecular sites as well as the hi& probability that only phenazine excitations have to be considered make this system ideal to try model calculationsusing the input valuesgiven in table 4. Without calculation the theoretical result of section 4.4 permits an importnat conclusion concsrning the lifetime of the triplet excitations responsible for the optical nuclear polarization. The large polarizations observed at room temperature require that the ratio of the relaxation rates with respect to the decay rate wJk (23) must be of the order of one or smaller. Since ihe relaxation rates among the triplet state sublevelsat room temperature are expected to be roughly in the range of IO8to 106s-l a correspondingly short lifetime is necessary for the triplet excitations reqonsible for the ONP. In particular, excited triplet states of traps or impurity molecules are inefficient as an origin of the ONP because their lifetimes are usually above the /Js-rangeeven at room temperature and their triplet sublevelpopulations are thermalized becauseof w,jk * 1. 5.5 Numerical results for the phenmine crystal Ihe return rates R, (2a) and the polarization rate (4a) have been calculated for an optical excitation cycle in the phenazine crystal and for a set of field orientations in the UCplane. Re two-electron (S = 1) one-proton (I = l/2) model has been used for the Q and /3proton positions separately, since the ONP result was shown to be additive fot suftkientiy

large fields. Although the spin density of the CH fragment in the u position is nearly four times that in the fl position their A, components are of comparable magnitude (see table 4). Hence the ONP contribution from both proton positions can be comparable, in particular for the orientation region of the maximum experimental effect with the field H about parallel to the moleculary axis. Consideringthe numerous parameters in the theoretical model, it is important to notice that most of them can be fixed from experimental knowledgeor physical considerations.The parameters of the magnetic interactions in the hamiltonian (6), as given in table 4 for phenazine triplet states localized in suitable host matrices, can be used in good approximation for all possiblephenazine excitations in the pure ehenazine crystal. Concerningtheir kinetic parameters, the values are more subject to details of the expected optical excitation cycle in the phenazine Crystal.Summarizingthe experimental knowledge, the following restrictions can be made quite general: (1) The relatively broad absorption edge at rather low excitation energy assuresan efficient and homo-

geneous excitation of phenazine singlet states in the crystal. Therefore, a triplet state population. even of most guest molecules [ 141, occurs via an intersystem crossingprocess in a phenazine molecule which is well establishedto be selective for the c-spin level (table 5). (2) Similarly,the decay is selective with respect to the T!j-levelif a phenazine intersystem crossingprocess is responsible.However,the selectivity in the decay seemsto be slightly smaller as compared to the population and this is not expected to change with an increase of the nonradiative decay at higher temperatures. Otter decay paths are also possible,e.g., a phenazine triplet excitation can be transferred to a trap where the population is thermalized prior to the decay due to a relatively long lifetime. Such a decay would correspond to a nonselectivedecay (tiN= 0.33 in (20)). Bimoleculardecay processes,however, should not be important for the initial population rates calculated here, neverthelessthe decay parameters allow a variation within certain limits. (3) The highest degree of uncertainty concerns the relaxation parameters. Here we restrict ourselves to the relaxation model of section 4.1 and the electronic zero field relaxation rate. u& is assumedto

D. StebD et aL/ONP in molecular crystals

Hp=800G

x

Y z

I

-3.8 -8.0

.OK .33 a25 J.3

Fig. 4. Computed polarization rate per created triplet state *O> as a function of the field orientation like in f&. 3. The hyperfine tensor used is that for a CH fragment in the Qposition of the phenazine molecule without nearest neighbour interaction.

be predominant corresponding to the relaxation mechanismof section 4.2, i.e., relaxation by a time dependent fine structure as a consequence of triplet exciton hopping among inequivalent molecular sites. In this case, the main open parameter in the model is the ratio of the w2 with respect to the other zero field rates w!& and w;~. A series of computer results is presented in figs. 4-8. The conclusions may be divided into the following points: A. Dpe of orientation dependence A quick glance over all of the calculated curves as a function of the orientation shows that the sharpest orientation dependence is found in an orientation range of about 50’ centered at 0 = -65’. As mentioned before, theyaxes of the phenazine molecules are then nearly parallel to the field. The directional

Y z

-I 0 -3.1 - 2.1

95 33 025 ..33 025 33

Fig. 5. Same as fii. 4 but with the hypertine tensor of a CH lragment in the @position of phcnazine and pointing along the long in plane x-axis.

cosines (16) of the field with respect to the molecular axes arep = 0.149, q = -0.979, r= -0.142. Most remarkable in figs.4 and 5 is the finding that the relaxation enhancer this effect. For fast relaxation the only strong polarization is found in a narrow range around B 5 -65”. 8. Amplitude wzd sign of the polarization As expected, the size of polarization depends

strongly on the difference of the selectivity in the population and decay rates. As demonstrated in fig. 6, the largest polarizations are observed if population and decay are selective with respect to different zero field levels.The polarization decreases when the selectivity in population and decay approach each other and changessign when the predominant selectivity changes from the population to the decay process or vice versa. The overall orientation dependence is unaffected by these changes in the selective population and decay parameters. ‘Ws behaviour is also independent of,the special choice of the relaxa-

D. Swhlik et aLlONP

in molecuhr

crystals

I/a

Fig. 7. Same as fg. 5 curve (C)

t-l .60 48I.kx’.33 .33 ,025 .20 .95 06 I.k' .025 06 20 33 ,. 5

but for a set of magnetic

fields

HP.

tion rate and of the hyperfme tensor made in fig.6. Even for an unselective decay process the highest polarization rate&(O) per created triplet state reaches the lob3 level (iIg.4) depending also on the field. If population and decay are selective with respect to different zero field levels, the peak polatition rates are coming close to the lo-* level. C lke influence of :he hyperfine tensor The amplitude of the polarization increaseswith the hyperfme tensor elements zleexpected from the increased rnlxlngcoefficients& It is interesting to note that the experimental observation of a positive peak polarization can be obtained only with a dominant $,-component In the hyperfiie tensor. D.

Fig. 6. Same as tig, 5 cum

(C) but for a set of sclectivcpopu-

Iattonand decay parameters01Indicated.

Influence of relaxation The selective relaxation w&B w&, w& reduces

primarily the polarization outside the range around e = -65’. Intermediate relaxation rates around w&/k * 10 can even Increase the peak polarization

D. Stehlfi et al jONP in nwlmtbr qystnls

183

sidered in all cases. In the phenazins crystal cross rc~dkuhlted laxation can reduce the peak polr GSt here. The influence is, howcvcr. 11 electronic relaxation occur. Effective cross relaxation proce~ &lIC .c in the if electronic energy levels approach el. ‘%rossing” of “level anticrossing” region. The ONP mechanism predicts here large nuclear polarizations due to the increasedmixing coeffkierlts e$. but they are destroyed again by cross relaxation which becomes very efficient in this region. In other field rangesspecitic cross relaxation processescan enhance and produce nuclear polar& tion effects 1221 and are expected to be primarily responsiblefor the experimentally observed effects in doped crystals. Further details are postponed to a subsequent paper.

6. Concluding

Fig. 8. Peak polarization rate in fig. 7 plotted 2s a function magneticfield HP and a parameter set of the ratio w&/k.

of the

remarks

Theexperimental ONP results in pure phenvine crystals can be reproduced rather accurately with the mechanismoutlined in this paper. The extensive knowledge of the parameters characterizingthe phenazine triplet state provides the favourablecondition that only one main open parameter w&/k has to be introduced in the model. Experiment and theory are in striking agreement if fast relaxation, w&/k about 100, and a predominant A,,,,somponent in the hyperfiie tensor are assumed.It seems too premature to speculate on this result. Only, one remark concerning the

(figs.4 and 5). When r&/k + 00the polarization ap: proaches zero (see section 4.4) even for IV!&,= w& = 0. With unselective relaxation rates the zero limit is reached faster, of course. Another strong influence of the relaxation is found tithe polarization field is varied. Whilethe orientation dependence is nearly the same with sufficiently fast relaxation (tig. 7), it is strongly field dependent without relaxation in contrast to the experimental result. In fig. 8, finally, the peak polarization is plotted as a function of the polarizing field with various relaxation parameters. Above a certain relaxation rate the field dependence exhibits a maximum as observed experimentally and it shifts to higher fields with increasing relaxation rate. A field dependence of the relaxation rate as given in (26) has not been introduced in the model yet.

A. 1. Spin functions for magnetic fields in an arbitrary direction md their rekttion to spinfinctims dffmcd with respect to a field in the z directin

E. Crossdaxation Cross relaxation of the case a in fig.2 mud be con-

All the molecules under consideration in this paper have in their triplet states physical quantities

hyperfine tensor is appropriate. The result does not

mean a predominant ONPeffect from the Bprotons, but rather that a time averagedhypertine tensor is likely to be responsiblefor those mixing caefficients cfiywhich are of primary importance in the optical excitation cycle.

Appendix

such as 13,E, A,, A, @ndA,, which are defined with respect to a symmetry determined molec~e fixed frame of refereoce. The plane of the molecule is defmed to be the x, y plane. ‘k%elong axis is chosen as the *-axis, the short a& the y-axis. We now consider a magnetic field H with comportents H.+,Z$ and Hz. Wecall the direction of this field the L direction. It is the effect of’ the strength and the direction of this field (the polarizing field) on the ONPwhich is the subject of our studies. We d$ke the followingset of direction cosinesp = HJ& q = H~~~~r = HJH, in which H is the field strength. The par~ete~ p, q and t ~interretatedby p2 + q* t rz J 1) determine the Z’direction with respect to the x, y, 2 frame. Wecan define a x’,yr plane pe~en~cu~~ to zi. From symmetry or any purely physical consideration there is no re+~on,however, to fix a particular direction as a coordinate axisx’ or y’. Wewill see, however, that there is a de~~t~o~ of x’ and y’ which is mathematt. c&y more &onve~entthan any other one. In our calculationswe diagonalizedthe spin hamiltonian as givenin the molecule futed x,y, z frame with spin function a and 8. Spin polarization, however, is more conveniently described with respect to the z’ axis, We therefore transformed afterwards the nuclear spin functions a and p to the x’,y’, z’ adapted spin onerous a’ and 8’. In what followswe give some of-the details of the transformations from a and B functions to a’ and 0 functions. A free proton in a magnetic field is described by a spin hamiltonian -gJ,H*Z (IO). We consider onlythe part Zf*Z=--&lx--$I"- fi,f,. Usingthe weit known results of the operators I# = X,y, z) on either a Or~Onee~~~er~~e~~at I-z*H)or=-~H,a.-i~~a-fH,P

Usingthe standard methods for solvingsecular eqwtions one fmds that the matrix (A.l) has eigenvaluesh =i4:f H. With a magnetic field in the z direction (p = q = 0; r = 1) ana proton has an eigenvalue -$I& and a fl proton an eigenv~~ +,&#,.By analogy one givesthe ejgen~n~t~onof -Z=ZZw&be~genv~ue -$ the symbol d and the ~~~tiou with eigenvalue t$ the symbol 0’. From the eigenvaluesof the matrix (A.11one finds a’= [(I +r}a+(p+iq)/3]2-L’2(1

+I)-“*, 64.2)

$= ~~~-~~~~~~

+@3jP2(1

+ry+=.

The phase factor of the eigenfunctionshas been

chosen such that for a field in the z direction a" = and@‘=& We defme further the two-spinunctions 7-2s=,(1.1~)[d(l)~r~2)+P’(l)oii2)l

a?

,

(A.35

These dei’iitions are completely analogousto those of TX,7””and T, in terms of a and P functions. Sub. stituting the expressions(AZ!) in fA.3) givesafter some algebraicm~~p~ation the follo~ug results:

From eqs. (A-4) it follows that for a field in the z direction TX*= TX;Trt = 7) and T,J = Tz. The phase factors chosen in the expression(A.2) lead to this . convenientbehaviour of the itnctions T,I, TV*and 7.~ for fields in the z direction. The physie~y convenient choice of phase factors leads to an unambigious choice for the t~sfo~ations from the xy,z frame to a x’y’$ frame. We can use the tra~fo~~ * tions fA.2) to find the transfa~at~on properties of the ~~ffo~s Q~&IJ, $$& and 2-r’2 &&~ + ~&~~~] = Tz*.Substitution of (A-2) in these expressionsgives

0.

afaI

Stehl&et al./ONP in molecular ayrk~ls

Comparisonof eq. (A-9) with eq. (k7) leads to the relations

-0 +r) 2

cL = [cQl t p’$ __oot(L+_@_iq)2-i/a&, - @ - iq)2 2(1 tr)

cLf = [-f&(p

T,#= (-p t iq)2_“2 QQ+ (p + iq)2-“a p/3t ‘Tz . M-9 A.2. Derivation of formulae (22) and (231 for the rate of decay from a triplet state TOthe ground state

Suppose we have an electron-spinnuclear spin state i with a spin wavefunction

For the decay of a triplet state to the singlet ground state one has to consider the selection rules for singlet triplet crossing [4]. These rules are based upon symmetry considerations for the electron spins describedin the molecular frame work; TX,Tw,T, are here the most convenient functions. For the nuclear spin polarization, however, the z’ direction is of primary importance and a’ and 0’ are the natural choice for the proton spin functions in the singlet ground state of molecules in a magnetic field in the z’ direction. We therefore choose a’ and 8 as basis functions for the proton spin in both the ground state and excited triplet state. Hence we have to transform eq.(Ad) to a slightly different form: G&,,= ~T,&cr’+c”&.

(A7)

From eq.(A.2) we obtain the inversetransformation ar=[(I tr)a’-(p+@)$]2-“2(1

tr).)-1’2, 64.8)

p= [(p -iq)d

+(1 +r)$] 2-“2(1 +r)-“2 .

Substitution of eq. (A.8) in e’q.(A.6) gives $&=

~T,“[&(l

tr)+&(p

- i@] 2-l”

x (1 t r)-1’2 t c T,B’[-&@ P +c$l

I85

tr)]2-*‘*(I

+r),)-“*

.

r)

t f&(p - iq)] 2-I’*(1 +r)-“’

t iq)

+&(I

tr)]

,

(AM) 2”‘2(I +f)-“2 .

From eq. (20) we obtain pie 7 X, k&i0 and $I= S, k&g in which do*and k’@l are the decay rates from excited state level i to the singlet ground gate _ with nuclear spin Q’and 0’ respectively,(k’ = k’i f J$n). The constant k,, is the hypothetical decay rate from a purely electronic level T,, to the singlet ground state (disregard@ the nuclear spin in all states). If one particular function TP is responsiblefor the intersystem crossingand we want to know the rate of crossing from a state Id to the ground state with nucIear spin function a’, we have to multiply k, with lc~l~. tt follows then that kid = l&gi2 and similarlytitic = lchl*. Substitution of(A.10) in these two expressions givesas a result the formula (22). For eq. (23) a completely analogoustype of derivation can be given.The basis functions for the electron spin are TX,Tu and Tz as used for the derivation of eq. (22). For the nuclear spins the field adapted iunctions a’&‘, $0 and Tz are the most convenient ones. Consequently we need the inverseof the transformations (AS) for the derivation of eq.(23).

References [ I] a. C. Maier, U. Haeberlen, H.C. Wolf and K.H. Hawser. Phys. Letters 253 (1967) 384. b. G. M&r and H.C. Wolf, 2. Naturfschg. 23a (1968) 1068. (21 a Ii. Schuch, D. Stehlik and K.H. Hawser, 2. Naturfrh~ 26a (1970) 1944. b.P. Lau, D. Stchlik and K.H. Hausm, I. Maw Res. 15 (1974) 270. [ 31 A.H. Maki and J.U. von Schiitz, Chcm. l’hys Lstten I1 (1971) 93. 14) J.P. Colpa, K.H. Hauser and D. Steblik, 2. Naacfschg. 26a (1971) 1792. IS) J.P. Colpa and D. Stchlik, 2. Natwfschg. 27a(1972) _ 1695. I6I D. Haarer and H.C. Wolf, MoI. Cryst. 10 (1970) 359. 171 R. Schmidbmger and H.C. Wolf, Chem. Phys tetters 25 (1974) 185.

•t i7)

[ 81 k SUM,Phys.Rev. Bl(l970)

(A.9)

1716, see appendix D. 191 H. Kolb abd H.C Wolf, 2 Naturfschg. 27~ (1972) 51. (IO] H. Haken and H. Strobl, 2. Phys. 262 (L973) 135.

186

D. Stehlik et aL/ONP in motecubr crystals.

[ 111 a N. Blocmkrgcn, S. Shapiro, P.S. Persban and 1.0. Mman, Phys. Rev. 114 (1959) 445. b.P.S. Per&an, phys Rev. 117 (1960) 109. c. WAC. Grant, Phys. Rev. 134 (1964) 1554. [ 121 kL Kwiram, MTP intern. Rev. Science, Phys. Chum. 3 (1972) 271-316. .[13] 3.A. Aniheunis, I. Schmidt and J.H. Van du WaaIs, Mol.Phys. 27 (1974) 1521. 1141 J. Cramer. H. Siil and H.C. WoLf,Chcm. Phys. Letters 12 (1972) 574. [ 151 Ii. Sixl and H.C. Wolf, 2. Naturfschg. 27a (1972) 198. [16] D. Stehlik, El. Haas and H. Zimmermann. Pmc Vlth Mol. Crystal Symp. Ehnau 1973, to be pub!sihed. [I71 J.Fh. Crivet.Chem. Pbys. Letters 11 (1971) 267; 4 (1969) 104; 3 (1969)445. [IS] V. Zimm ermsnn. Thesis. Stuttgart (1969).

[19] D. Schwcitzez and I. Bchnke, unpublished results. In ref. [18] lluorenc X-trapsnext to a diinzothiophenc guest mokule wae olnerved and the hype&e tensor of the two CH, protom is found to be different. In contrast. the fluorcne X-trap in an undoped crystal of fluorene-d,& at 4.2 K show-san equivaknt and nearly isotropic hypcrfme tensor with an isotropic coupling constant of a = 20.5 MHZ(= 7.3 gauss) for both CH, protons. [ 201 J.P. Colpa and S.KesmarQ, unpublished results. 1211 CA. Hutchison. J.V. Nicholas and G.W. Scott, I. Chcm. Phys. 43 (1970) 1906. I221 a P. Lau, D. Stehlik and K.H. Hauscr, Rot. 18th Congr. Amp&, Nottingham (1974). b. D. Stehlik, Rot. 18th Congo.Amp&c, Nottingham (1974).