Calculations of optical nuclear spin polarizations for randomly oriented molecules

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final form 13 May 1985 1

Opricd spin pokizations ha& been ulculnrcd for qstcms with mndomly oriented rn&cula (&sses. poIya-pmIs)_ Bi this mcchjnism polarintion~ &rise uadei opticatexitat& d& to sekctiviity of inknokcular intersystem crossings One-nudcx mokcuIa haw been co&id&d_ Electron and nucku pokuizaddns have .bccn cahxdatcd numuicaIly f&r ee ground_+iqIer and the ardttd tripk stateik has been shown that in crystalsand in systems with tandomiy oriented moIecuIes_ the scak of ekctron pokwizations is in fact the same. while nuclear polarization eff&ts arc lower by ~-6 orders of magnitude in the Iatra

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czasc

1. Introduction By optical spin polarizations one usually means non-equilibrium spin populations of mokcular~ sublevels arising under photoexcitation. This phenomenon has been adequateiy studied experimentally and theoretically for molecular crystals of aromatic compounds (see, e.g. refs. [1.2]). The physical nature of the effects under discussion is as follows: Having absorbed a photon, a molecule becomes singlet excited_ It might thkn pass~.to. either the groind sir&et state S, (fluorescence, radiationless decay), or the tkplet state .T hue to ik.ramolecular I intersystem cro&ing induced by spin-orbital &upl~~g_~Th~ dipolar coupling of tripiet &e&on spins spli&_ $is‘ state (the so-&Ile$ zero-field splitting, ZFS) into &blevels with- vatiT ous symm&ries_ As a -result, -they are differetitly mixed w&h the singlet st+es by spin-orbital coupling. The r&uiting sele&ty ‘of intrarnolec@ar ~transitions leads to differing -populationsof the t.ripIet
~_

._- :

to the field direction. This phenomkoe was fikt .. discovered .by Schwoerer-and Wolf in 1966. for. organic compounds_ A further progress -in this field is associated with investigations b$ .van der .Waals, Hausser, .de Gfoot, Schmidt El-Sa\;rd~etc__ ~~ (seethe detailed review [I])_ The intersystem crossings induced by spikerbital coupling are purely electronic_ Ijowever, the hyperfine coupling of electrons with magnetic nuclei might -dso result in polarizatioti of thenuclei through selective transitions; This mechanism of optical nu&ar spin polar&ion has b&n first studied theoreticaJly _by Colpa et al_ [3]. _.- : The spin polarikio~ mechanism understudy is. characterized by the pronounced dependen& of the magnitude and the sign of the effect on ihe triplet molecule orientation relative to.the exyemal: magnetic field dkc!ion. This ‘dependence stems from a strong anisotropy of dipolar coupling be-~ tween the unpaired electron_spins. The _qu&on L arises whether spin polarizations car.-follo~~%is ~-1 ktec&nisin in systems -.wif” =randbml$ .orienfed This qu&ti&l+. I:molecule&(glasses, po!ycr+k)_

-._0301-0104/S!i/~O3~30 $3 Elsevier Science Pubiishcx$ B-G_ (North-Holland physics @blishing Divjsion) .. _.-: : ~, : __-_ ‘; :_-

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been partIy &vered by wong et aI_ [4] who have. demonstrated the eIectr& polarization averaged over aII molecule ~orientations td be non-zero_ SFehIik 121believes that nuclear polarizatio& may. in principle arise in gIasses under opticai pumping_ However. the nuclear polarization effects in the abovementioned systems have never been

calculatedThe aim OF Fhe present work is to calculate spin

nel in the overaIl transition rate. Ci,pi; = 1);. &ii1 the rate constant of transitions from the ith T

state to the nuclear sublevel I of the groutih state S, (K, is the overall rate constant of T --, S, iranSitioxis_ X-, the ‘sPgific -*eight of the’ Ti -+ !$,, channel in the overa!I transitioti rate. &,k, = 1). AI1 the processes can be-described Iowing system of balance equations:

by the foI-

polarizations in molecuIesr oriented randomIy refative to the externai magnetic field, arising under optical excitation due to selective intramoIecuIar intersystem crossings-

forn& kinetics of the pmcesses resuhing in spin poIa&aCons

t The

Let usconsider a one-nuclear molecule with a spin of l/Z_ Fig_ 1 show5 the processes responsible for nuclear polarizations arising under photoexcitation_ Le: re,(l= a. 8) be nuclear spin populations of the ground state S, subIeveIs_ Hi( i = a. #?) be the sublevel populations of the excited singIet state SI_ In our case the indexes a, B conform to

states

with

different

projections

(%1/Z)

of the

nuclear spin j onto the quantizadon axis z. The rates of transitions between the states under discussion are defined as: It;_ the optical excitation rate constant for the S, state: Q_ the S, decay rate constant: P-pi, the rate constant of transitions from the jth S, state to the ith T state (PO is the overali rate constant of S, - T transitions_ pii the specific weight of the S,, - ?;: chan-

7E i -

i

(i=l

._._.6).

The nuclear spin (?) and the overa electron spin (3) projections onto the e..temal magnetic field direction determined by the unit vector n obey the folIowing expressions: in the ground state

ms =OL -

(4)

q3)r-;

in the triplet state (J,),=Ci~(ij(ni)

(5)

Ii).

i

Here ii> is the wtivefunction of the ith stationary triplet subleveL Since the intersystem crossing induced by spin-orbital coupling is purely electronic. !he overa rate constants. of transitions from the nuclear spin sublevels S, to -ail- the triplet States must be e+uaI, i-e_ Xi p,- = Eipsi = l/2. Similarly. .&ik, = Eiikii = l/2_ Hence. eq_ (2) Can+ tititien’ : as dH,/dr=

Wnj-QHj-(P~)Hj_

(7)

I Fig1_scbcnuofpnxrssa l#d.er lzxciucio~

responsible for spin polarizations

Let us determine the stationary IjopuIations for the s- and T-state subleirels under co&ant opti-._ Cal pumping Substituting Hi from &_ (7) to eq:

" :- ....

" . . +: :

.

. :-: S-A-+Sul~enkO.+IC~£.Sat'dcbot':l.Opt!ealnhde, m':spinpola~-ati~":

+(3). o n e o b t a i n s t h e s e t o f . - e q u a t i o n s :: :: i+:+.+ - : "+ "

: [:+.j(++ + _ [2$VPJ(2Q

++ P o ) ] ~-'-Pid+j : Kok+tv+ = 0 (

_

-:--(9)

+

+ J • . + w h e r e L', =-- ~ k + j . ".- - : -- " -. --- : : " ". T h e n u c l e a r p o l a i i z a t i o n in t h e g r o u n d " s t a t e c a l c u l a t e d p e r m o l e c u l e . i s = 1 0 - ~ - 1 0 - 3 [2]. i.e. n o a n d n a d i f f e r byl a q u a n t i t y s m a l l e r t h a n 107 ~, n,, -----h a - - - - 1 / 9_ H e n c e . t h e s t a t i o n a r y p o p u l a t i o n o f t r i p l e t s u b l e v e l s t a n b e c a l c u l a t e d in a first a p p r o x i m a t i o n b y eqs_ (9) w i t h n , = n # = 1 / 2 . A s a result. Po)] (

pJki)-

(10)

w h e r e Pi = ~.jPiiS u b s t i t u t i n g e q . (10) i n t o eq. (8), w e h a v e x~ith t h e same accuracy

"i = ~ ( P,/k,)k,i-

(11)

i

S u b s t i t u t i n g t h e e x p r e s s i o n s o b t a i n e d f o r s p i n level p o p u l a t i o n s t o eqs_ ( 4 ) - ( 6 ) . w e o b t a i n

(i.>~ =

• Y'. ( p + / k , ) (~,.- - &:,#).

(12)

Here the first rightrhand-side +erm describes~ the-Z e e m a n i n t e r a c t i o n o f t h e Overall e l e c t r o n s p i n - S + with the external magnetic field H-(~is the--~ : tenstJr), 'flae- S e c o n d t e r m i s - t f i e n u e l e a r : - Z e e m a n interacti0n~ the third term is-the'el0=~n;dipola]r-.c o u p l i n g (D is t h e Z F S t e n s o r ) , t h e i a s t - t e r m i s t h e . hyperfine interaction (hfi) of the overait-ele~ffon ~ p i n w i t h a - n u c l e a r s p i n ( A i s . t h e hfi i e n s o r ) . . T h e ~ - t e n s o r a n i s o t r o p y is s m a l l f o r m o s t o f t h e organic triplet, molecules. For real systems very often the prindpal axes of the tensors b and A-do n o t c o i n c i d e . H o w e v e r . it s e e m s t h a t t h i s n o n coincidence of tenso r axes affects very little ONP f e a t u r e s a n d it c a n b e t a k e n i n t o a c c o u n t i f o n e substitutes principal values of/3, by diagonal-ele- ments of A in the D-tensor principal axis system [9_.5]. K e e p i n g t h i s i n m i n d a n d tO s i m p l i f y : '._he calculations we assume that principal axes of the t e n s o r s b a n d ~, coincide_ I n t h i s e a s e . - t h e s p i n h a m i l t o n i a n (15) is in t h i s m o l e c u l a r c o o r d i n a t e system " -

+

= q~'~- ( P + / k + ) ( i l ( n i

) !i)-

(13)

q~-'~ ( p , / k , ) ( i l( n S ) l i ) .

(l 4)

i ('~.)r

=

i

where q= WP, JKo(2Q+ Po)- I n n o r m a l c o n d i t i o n s q << 1 w h i c h c o r r e s p o n d s t o a l o w c o n c e n t r a tion of excited triplet molecules in the sample. T h e q u a n t i t i e s kij and Psi cha.caeterize t h e eff i c i e n c y o f v a r i o u s t r a n s i t i o n c h a n n e l s . T h e 9- d e p e n d o n p a r t i c u l a r - f o r m s o f i h e v,-avefunctions f o r triplet snblevels. Therefore iris necessary to determine the spin eigenstates of triplet molecules.

rate

crosmng

arbitrary magnetic field

-

.

+i:

constants -_

-- "

E+ A ,

-

(16)

_

w h e r e w,:. a, t a r e e l e c t r o n a n d n u c l e a r Z e e m a n f r e q u e n c i e s , r e s p e c t i v e l y , in a n e x t e r n a l m a g n e t i c f i e l d d i r e c t e d a c c o r d i n g t o t h e u n i t v e c t o r n;- D+ E are ZFS parameters; A i are the principal values of t h e hfi t e n s o r ( i = x , y , z ) . . . . . .. -. A s b a s e w a v e f u n c t i o n s in z e r o m a g n e t i c field, i t is c o n v e n i e n t t o u s e t h e f u n c t i o n s IT., V ) w h i c h a r e e x t e r n a l p r o d u c t s Of t h e e i g e n f u n c t i o n s f o r t h e ZFS operator (the third term in eq. (15)),.-ITs) ( p = x . y+ z ) a n d t h e e i g e n f u n c t i o n s o f t h e o p e r a t o r ] : , i.e. I V ) = l a ) , I f l ) - L e t u s e x p a n d t h e w a v e eigenfunctions Ii ) o f the spin h a m i l t o n i a n (16)

o

3.-Interso'stem

+ D(S__z - - $ 2 / 3 )

- . ~ ' r = ¢o,:( n . S ' ) - - c o , ( h i )

i (/.)-r

. u : ++. - -+-i:+,~++Y:~-;.~:++-~S:_~ .°-

~ h a m i l t o n i a n-. :. +:.? : - - : - . . -

::: i+) :+++++:.++++s: +,:Jr-,) + s+:,+++sAi/+'!::++++

..

Ni = [ IVPo/A%(2Q +

+ " :-: :':_++::+":+~j::Y'-~?::f~?~33,

in.the,

functionslT~.

V)....

+

.

.:

+

:

in

an :-+=

" ] i ) = ~

+ .p . v

'

" ' V

-'

R,~IT,.

) ..... - -

:

"

.

-

"

-

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+ . +

"

+ --

. -

.

-

(17) :i.-

_

. T h e w a v e f u n c t i o n s f o r triplet, s p i n s u b l e v e l s a r e w h e r e R'~v a r e c o e f f i c i e n t s ~d e t e r m i n e d +b y - t h e determinedby s o K q n g p r o b l e m s _ of_ e i g e n v a ! u e s S t r e n g t h a n d dLrection o f . t h e m a g n e t i c field . . . . "- . ~ and of-eigenfunctions for+a triplet molecule spi n T h e r a t e c o n s t a n t s o f d e f i n i t e i n t e r s y s t e m Cross-

--

inzs are proportional to the square&matrix -eIement of the spin-orbital coupling operator ti_ pi~=c,Ij’,

kij=C,I(iI~IJ;,)I’W)

Substituting expansion (17) into eqs_ (18) affords the following .equation reIating intersystem crossing rate constants in an arbitrary magn+tic field to corresponding matrix elements V in zero field:

~....

(IS). we have

pp=Iypsr~~. cp,= P

1.

-x;,+,TsI~. Ck,= P

I_

1 (22)

As a result. the reIationship between the constants and anaIoa,us quantities in zero field are

pii_ kij

pji=f~IR;jI’p,_ P

kij=;~IR;jI’kP_ P

(23)

4_ Calculated data

in investigations of optical nuclear polarizations (see, e-g_ ret [21). pii and kji are usually calculated neciecting interference terms of the type Rij( R’,i)*V,V;z by the expressions P;~ = $ c 1Rij,iVp” (=. P

k, = f c 1R;,Yp” P

1’

(21)

instead of eqs_ (19)_ In general. the possibility of using eqs. (21) instead of eqs_ (19) is an open question. However. there are evidences that eqs. (21) give quite reliable results [2_6.71_The problem under discussion has an anaI_yticaI solution in the approximation of a high magnetic field_ In this case_ the interference terms of eqs_ (19) vanish if the magnetic Field is directed along the main axis of the ZFS tensor_ When averaged over aI moIecuIar orientations. the contribution from these terms to the net polarization is aIso zero_ A good argument for eqs. (21) is the agreement of the data obtained with experiment [2]_ An attempt has been made [7] to determine the contribution from the interference terms to the intersystem crossing kinetics_ Experimental phosphorescence data have shown a better fit to cakuhitions by eqs_ (21) than by eqs. (19)_ Thus_ the constants pji and k, are expressible via the matrk elements squared. I V, 1’ (20) which can be determined experimentally in zero magnetic field [l]_ Let p,(p = _K?_t__I) be the specific weight of the St -T channel in the intersystem S, -T

crossing, X-, be the specific weight of the T - S, channel in a triplet decay_ As in the case of eq.

For mokcuies which are oriented arbitrarily relative to the magnetic field. the wave eigenfunctions cf the spin hamiltonian (16) can be determined only numerically. In the present investigation. we empIoyed standard methods to caiculate the coefficients for the eigenfuctions of the hamiltonian (16) expanded in base function at zero fidd for various strengths and directions of the external magnetic field_ Thereafter the intersystem crossing rate constants calculated by eqs_ (23) were substituted in the polarization equations (12)-(14). When integgated numerically over a11 angks. the above quantities gave corresponding electron and nuclear polarizations averaged over all molecular orientations for the ground triplet and singIet states_ The numerical integgtion was performed in the following way: At first we integrate over the azimuthal angles +_ At a @en value of the polar angle B for n values of 0 (with a step 2=/n) we calculate a vahte of the integrand (f) and then approximate the integral over + by a sum I,,= (25z/~z)~~._,f(~~)_ Then we double the number of steps and calculate Im_ This procedure was repeated until the following requirement was fulfilled

for c = 0_02_The smallest angle step wexsed was 2-E/1024_ Integration over- 8 was done using the Simpson approximation_ The number of steps was doubled until 3-456 accuracy was achieved:. Our calculations were performed for a number

of typical pammiters of organic triplet m&c&s.

ters in zero field:; D=GiG72 cm-‘;.E=‘-GBG8 cm+’ which &responds tq.anthrac%ne molecules The printipal values for the hfi tensor are A_; =. -24.8 MHz. A, ~? - 8.0 Mk ii; = - 16.6 MI-k It is assumed that in zerofield the only transition canoccur to the TX state, and-all the tripIet states decay with the same efficiency, ire_; p= = 1; pl_= p, =G. kx=k_,.=k,=1/3: Fig 2 (curves l-3) shows pIots of the triplet electron ..spin pohuizations versus the external magnetic field directed. along -the *es of the molecular coordinate system Curve 4 is the field dependence of electron polarizations averaged over all possible molecular orientations relative to the external field direction_ According to the OEP literature (9 eg_ ref_ [13) in the principal directions of the D tensor, OEP is small in comparison with that for the orientations of mokcules when direktion of external field does not coincide with b-tensor principal axes_ The averaged value of OEP (curve 4) is comparable with OEP in monocrystals for a magnetic field oriented along the principal axes of the 5 tensor (curves l-3). This is not the case for nuclear polarization_ Its field dependence for the same parameters of a triplet molecule is shown in fig 3 for the ground state and in fig_ 4 for the triplet state_ Cunes 1 and 2 pertain to specific magnetic field orientations relative to the molecular coordinate system. It is seen

Fig. 3_ Fidd dependenm Of nudear poh-hions for Lhe -~ ground state: (1) the field is died aIong the x axis (2) tic fieId is directed atong the z axis of the mokcular coordinate. system. :

that the nu+ar polarizations -noticeably differ-. from zero just within definite. sufficiently narrow ranges of magnetic field strengths. In these fieids, the proximity of the triplet spin sublevel energies has been shown 12.81 to be maximum (with hfi negiected, the triplet spin subleveis cross in these fields) which resuhs in matimum nuclear polarization effects The ONP has an alternating sign. In systems with randomly oriented molecules, the averaging of nuclear polarizations is the superimposing of the curves of the type shown in figs_ 3

< ix>, t

I

Of

<

-049,

2

I

I

i 2q

0

25

75

H.mT

FIN 2_ F&d dependenccs of triplet eksu-on polarizations in variausIy dinzctedmagnetic fiddsr (l-2.3) the ZieIdis directed abng the axis of the moIecnIar coordinate synem x. _v. 2. respuxiveI~ (4) poIariza~ionaveraged over all moIecuIar orientations. The mokadar parameters are giwn in the text_ -.

-. 25

75

H.mT

::

Fig 4. FieId.dependences of tipIet nudear poIari+ions
-

Ii/\--i 0

25

7s

H, atT

fib 5_ F&d dependence of nuctear polatitioions state or Gtndw oticntcd molccuier.

for the Smwtd

and 4_ For subensem bies of moIec&s with different orientations maximum polarizations are obsen-cd in different fields. As a result. the polarixations in such systems are to be much lower than those in crystaIs for optimum external field otientations Figs_ 5 and 6 show the field dependence of nuclear polarizations for the ground (fig_ 5) and triplet (fig 6) states averaged over all orientations When averaged, the polarization effect is seen to essentialiy reduce (approximately by two orders of magnitude) as compared to that in crystals with favorable orientations_ The calculations were done without taking into account -the spin relaxation effects Professor D_ Stehlik pointed out to us that spin relaxation can increase the average ONP in randomly oriented systems_ The relaxation broad-

ens the Iaro, ONP contributions in the l-eve1&tticrossing reggons -without reducing ’ t’le absolute polarization [Z.5]_. Hence.one expects’ that-relax& tion can ‘play an important role .for cJNp,in i-andomly oriented systems also. : -: I. Our calculations have thus demonstrated the. possibility of detecting. optical nuclear polarizations in experiments with randomly orienl’ed molecules (glas&_ ‘polycrystals)_~ The required .opticaI pumping must ensure the nucIear polartition, in a corresponding. crystak exceeding the l IquiIibtium Bohzmann polarization by not less than two orders of magnitude under favorable- orientations. This requirement scents. to be attainable in .rcal conditions_

AcknowiedSement The authors like to thank Professor RZ Sagdeev for stimulating discussions and Dr_ NV_ Shokfrirev for useful advice in numerical calculations_ We wish to express our sincere thanks to Professor D_ Stehlik for essential comments and useful discussions

References [II

i

4s

/\-1

I I II [!

2s

7s

I

H.mT

ES. 6. Fidd dgndtxtcc of nucl& pokGzzttions for the triplet state or Izndond~ oriented InotattIcr

K-H. Ha-

and H-C_ Wolf. Advan_ 51agn Ru

S (19?6)

85. 12) D_ Stddik. inz Excited state% VoL 3. cd_ EC_ Iim (Academic Press Nzw York 1977) p_ 203. [;I J.P_ Colpa_ K&i_ Haand D. Stchlik. Z_ Naturforsche 3>(1977) 652. [41 SK_ Wang. DA Hutchinson and J.K_S. Wm. J. Chctn Phyx 5s (1973) 9S5_ 151 V_ Macho_ J.P. Colpn and D_ Stehlik. Chem Ph_vs. 4;1 (1979) 113. [6] D. St&I& A_ Do&ring. J-P_ CoIpa_ E_ CalIqhan and S Kumarky_ Ch&n Pit>% 7 (1975) 16% [7] C-C_ F&x and S-L W&tnan_ Proc Natl. Actd. S&i. US 72 (1975) 4203. [SI J-P_ Colpa and D_ St&l&. Chetn Ph_vs_21 (1977) 273_

.-