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

Chemical Physics 21 (1977) 273-288 0 North-Holland Publishing Company

OPTICALNUCLEARPOLARIZATIONAS A CONSEQUENCE OF THE NON-CROSSING RULE (LEVEL-ANTI-CROSSING) I. Analytic;lltreatment of ONP in the level-crossingregion J.P. COLPA * and D. STEHLlK ** Max Planck Ihsfifute, Deparcrnenr of hfolccubr

Physsics, 6900 Heidelberg, FR G

Received 4 October 1976

The genera\ [armal theory for Optical Nuclcnr Poluizalion (ONP) through an optical cscitation cycle is worked out for the special case of ONP as a consequence of the non-crossi% rule (Lcvcl-AntiCrossing, LAC). In doped cryslals certain guest-host complexes in a triplet state play ;I crucial role; they have rather low 0 and E values and consequently their LAC regions are at much tower field strengths than one espccts for normal molecules in 5 tri+t state. The field and orientation dependence of ONP is treated anatytictily far a two-electron one-proton model by means of, perturbation theory. One fmds then that the field dependence is givenby ;I superposition of two bell&aped functions, opposite in sien and differing in centre and width. Under certain circumstancesthe ONI’CUNES become a superposition of two lore&an functions in the hi@field approximation. The widths and the dispance between the centrcs arc dctermirrd by the elements of the hyperfine tensor. The superposition pattern mentioned here is found esperimenlally in the strongest ONP effects measured so fx in doped crystals.

1. Introduction

For the benefit of the reader we summarize briefly

the essential elements of an ONP experiment. A molecOptical Nuclear Polarization (ONP) has been dip

covered in 1967 [1] and the phenomenon has been observed since in a number of pure and doped molecular crystals [2-.7) . One obtains with ONP remarkably large polarizations at room temperature. The highest polarization has beeri achieved in fluorene doped with acridine, using a polarizing field of 80 gauss together with W radiation. It corresponds to a Boltzmann polarization in an external field of 3 X lo6 gauss. A re. view paper, summarizing the ONP literature in the context of Optical Spin Polarization in general appeared recently [S] ; another review, discussing in particular the mechanism of ONP in molecular crystals and a qualitative analysis of the level-anticrossing effects (LAC) was published by one of us [9] _ On sabbatical leave from Department of Chemistry, Queen’s University, Kingston (Ontario), Canada. ** Present address:Freie Universitit Berlin. Fachbereich Physii (FB 201, Institut Atom und Fesikorperphysik WE lB, KG nigin-LuiseStrasse 342, 1 Bcrlin 33. l

ular crystal (pure or doped) is placed in a magnetic field (the polarizing field) of variable strength and orientation with respect to the crystalline axes. and is irradiated by W light. Some molecules (guest or host) are excited to the lowest triplet state, commonly via intersystem crossing from the lowest excited singlet state S, -+T, following singlet absorption S, + $. The decay T, + S, back to the ground state estabhshes an excitation cycle during which a nuclear polarization of the nuclear Zeemanlevelsin the ground state is generated. After the light is turned off the polarization is most conveniently measured by a two-field technique [I ,2) in a fixed measuring field. The polarization is defined by the population distribution over the nuclear Zeeman levels + l/2: P = 01, - n_y(rr+ + II_).

(1)

Usuallyit is more convenientand sufficient to measure the initial polarization rate j(0) starting with an unpolarized sample before light irradiation. p(O) is propor-

J.P. Coipa, D. Ste/rlik/ONP as 0 consequeme

274

tional to the number of polarized nuclear spins per created triplet state which is the quantity of interest in this paper. In search of ONP mechanism an explanation [la] analogous to the Overhauser effect failed to account for csscntial experimental observations. Instead a mechanism [IO] was proposed with two main features: , (1) Selective populalion and decay of the electronic triplet spin states during intersystem crossing. (3) Mixing of the electronic and nuclear spin states as a consequence of the static hyperfine coupling during the liretime of the excited triplet state. The latter enables the nuclei to participate in the spin selectivity although this is of a purely electronic origin. This mechanism has been worked out to a mode: for ONP as a consequence of an optical excitation cycle [I 1,121. The agreement with many experimental results is very good. Already in ref. [IO] it has been pointed out that particularly large ONP is predicted for field strengths and orientations “where one expects a crossing of the energy levels in case one neglects the hypcrfrne interaction. This interaction gives rise to a non-crossing rule &AC) for the magnetic substates . . ..” In other words: LAC by hypcrfne interaclion was suggested to cause very large ONP. In fact, LAC effects in the electronic triplet spin states have been optically detected via phosphorescence in many esamples [ 131 as reviewed IS] in the context of ONP. On the other hand experimental evidence for the predicted ONP by LAC was not available. For the known molecular triplet states in the pure and doped crystals with large ONP (see table 5 of 191) one calculates level crossing fields between about 300 G and 1100 G, while the most prominent ONP efrects in doped fluorene crystals occur at considerably lower fields around 100 G. It has been suggested already in [IO] that relaxation, in particular cross relaxation can destroy the large polarization predicted in the LAC region of the known triplet states. In order to explain the observed large ONP at low fields by LAC, the possibility remained to postulate new triplet states with the appropriate molecular properties, in particular a signiJicantly smaller zero field splitting. This concept [9] has been based on solid evidence by the direct EPR detection ofsuchtriplet states in fluorene crystals doped with acridine or phenazine [ 141 . Besides the EPR lines of the isolated guest triplet

an

of the

non-crossirrgnrle. I

states one detects in these spectra additional lines due to shortlived triplet states with a considerably reduced zero field splittjng. They are assigned to the triplet state of guest-host heteroexcimers. For the acridine-fluorene complex the EPR measurements yielded IDl/Ir = 396 MHz and IEI/Ir = 27.9 MHz with the corresponding level crossing fields H& = 141 C andHiC = 55 C. Precisely around these tields one observes the largest nuclear polarizations associated with a characteristic field dependence as shown in fig. 2 of part III of this series [7a]. The ONP is plotted as a function of the field oriented for maximum ONP at low fields. These orientations can be assigned to the principal axes of the zerofield splitting tensor of the complexes. Hence the earlier idea of ONP by LAC [IO] as contained in the model described in [12] leads to a basic understanding orthe ONP results in low fields [9] LIn addition it becomes possible to use ONP as an independent spectroscopic method in order to determine the molecular properties of the triplet state complexes. With this aim in mind a quantitative understanding of the ONP results should be achieved. The present papers are hoped to be a step in this direction. In this first paper it is shown that the ONP in the level crossing region can be described analytically with a perturbation theory treatment. The basic features of ONPby LAC can thus be interpreted in a most direct way. The second paper deals with the crucial effect of the relaxation on the field dependence of the ONP. This will require the use of the general model as outlined in [ 1?I This model is also necessary to interpret the ONP r&Its with the field oriented along the third principal axis of the fine-structure tensor, the noncrossing case. In this context we note that the ONP results obtained in pure phenazine crystals and their interpretation [12] represent essentially this noncrossing case. The most prominent polarjzation effects were observed with the field orientation such that the molecularyaxes of the phenazine molecules are aligned as near as possible along the polarizing field; for phenazine molecules they direction represents the non-crossing case for the purely electronic spin Iunctions. Moreover, the importance of selective electronic relaxation has been clearly demonstrated for ONP in phenazine crystals. Selective electronic relaxation rates can enhance the ONP for a particular orientation and magnetic field

range whereas elsewhere the relaxation destroys the polarization as expected in general.

2. Level-,4ntiCrossing hyperfine interaction

in molecule triplet statesdue to (polarizing field in the z direction)

Althou& we will later use computer calculations for a comparison between measured and calculated ONP, it appears that many of the basic concepts can be obtained by a perturbation theory treatment and that under certain conditions analytical expressions can be derived for ONP as a function uf field strength. These expressions will serve as a physical background for the interpretation of experimental results and their comparison with more elaborate computer calculations. At the present stage it seems justified to treat only the basic principles and to avoid any unnecessary complication. The essential features can be treated by a twoelectron one-proton model under the assumption that the electrons are coupled to a triplet state and that the principaI axes systems of the fine structure tensor and the hyperfine tensor are identical. We treat first a case with the polarizing field in the z direction. The notation and nomenclature will be as much as possible identical to those used in ref. [12]. We neglect the usually small anisotropy in theg tensor and useg=gx, =gyy=gzz for the electronic Landi factor. Hence we consider a system with the spin hamiltonian [14] s1= D(S2Z - 4s’) t E(SJ.r - S’) H 1'+ @S.?I

Table 1 Spin hamiltonian with zero-field electron-spin functions 3s basis functions for the subspaces 1~and II:, respectively. The function T.,,is used in the comples definition [ 151asdone also in [ II] and [ 121,such in contrast to ref. [IO].In thcsc matrices and all similar ones, only the upper triangular part is given. The lower part can be found by the relation f$ = H;i

--

--

tz T,(e) 0) _--p-D - &-- &,&H,

TJ(c) 01

T:(c) P(n)

-i&l=

&lYr

+g$Y,)

D + E - &,&,Hq

In table 2 of [ 121 we have given this spin hamilto-

nian in matrix form under more general conditions. For a field in the z direction we get the direction cosinesp=O,q=O,r=l intnble2of [12].Thematrix blocks out in two 3 X 3 matrices which are given in table 1; this feature is a matter ofsymmetly, not of any approximation. We refer to the basis set r,(e) a(n), TY(e) Nn), TZ(e) o(n) and any linear combination of these three functions by an index Iz; index 1I.zrefers to T,(e) p(n), T&e) 0(n), T,(e) a(n) or any linear corn. bination of these latter functions. One notices that one can derive the matrix for subspace IL from the matrix

iI y’r

--Ik

T,W

P(n) ----

--

Ty(4

p(n)

_---~_---_

Tzk)

dn)

--

for subspace Lzby changing the signs of g,, , A,,,,,and A,;. We have shifted the absolute values of th’e’U@ona\ elements in table I such that the pure electronic function T_ has zero energy in a field parallel to the z asis. When bne neglects the terms involving th? nuclear spin. and A,? one gets a purely electronic i.e.,gn8n,A,,n,7 problem for which the secular equations are easy to solve [15]. One finds that the linear combination of T,(e) and Tr(e) with the lower energy crosses the T,(e) state at a tield [9] : I-fir = (l/g@ JKZ

(2)

-ii

~~$,&

W

At the field Htc one finds for the coefficients qY and CY of Tr and T), the following results: For the upper state u: c”, = -d_

and CT,= -id-

and for the lower energy state II: C8,=d@qDj

and C_;= -id-. Ub)

In most cases and in particular in the complexesmentioned in the introduction one has E Q D. The coefficients just mentioned get then absolute values -j-Ill , and the resulting electronic spin functions become the ordinary high-field functions: I+)= I&Y)= -2-“2

(TX t iTj,>

J.P. Colpa, D. SteldihjONPas a consequence of the non-crosxiq nrle. I

276 Trible 1. Spin hamiltonian

with hi;h field basis functions -___

for subspaccs k and 1k

(112d%4xx

D - &,P,H, - iA,,

+ A,.,,)

- M, k&Hz

---

----IIZ

I+) P(n) ----

I-)P(n)

Ti a(n)

D + :PnPJfz I - sffzz + ml,

E

(1/&&(A,

+ AYY1

and

negligibly small and we get in good approximation:

I-) = Iflj, = 2-l” (7; -- iT,>,

- H& HL' LX

in good approximation already at H&. We will call this the high-field approach. When E
We take the LAC centres as coordinate centres for the fields and define:

f& =CD- 1QlkP +Q”>,

(4a)

&

(4b)

= (D + I~;,)I~:P

For protonsgP/g,&

-Q,).

= 658. Hence theg#,,

term is

= 4,fg~.

(4c)

AH, =Hz - HFAc,

(54

MIlz = Hz - HzC.

@bj

We substitute (Sa) and (5b) in the matrices for the subspaces Lz.and IIz respectively and obtain the results summarized in table 3. The quantities F,and Fu in table 3 are defined by

Frr = -i@

+ ~~,knP,/kD

- R&J-

These quantities are small; they can be subtracted from the diagonal elements, this means a minor shift in the zero point energy. Moreover they have the same coefficient for the two sets of anti-crossing levels and do not affect the energy difference in the LAC situation.

217

J.p. Colpc~,D. Sfelrlik/ffNP PSQ conseqrrence ofrhe norI-cmssiflg ndc F Table 3 The spin hamiltonian [or the casesLZand IIZwith the LAC centres -k

I+) a(n)

&-

coordinatecmtrcs lor ihc field strength

I -) dn) + 20 - 35

&7,)ti~ -

-----

as

--

---

f E _(_____

T, P(n)

_______

; -(gP + Ig,P*)4H~

I_1J.z

I +) P(n)

+ FI

I I

I -) Pm -

So far the treatment of the spin hamiltonian (2) has and table 3 gives (2) in convenient matrix form to investigate the LAC effect. Some simplifying approximations are justified and used now for working out the results in the region around WHi’,“= 0: (a) We assume that the high-energy spin wave functions arc given by I+) a(n) and I+) /I(n) respectively and that in the LAC region the wavefunctions are h-t= ear combinations of I-) a(n) with T, P(n) for case 1 and of I-1 p(n) with T, or(n) for case II. A more de. tailed perturbation treatment shows that in the LAC region the degeneracy or near degeneracy of the lower two levels gives by far the most important effect; the influence of the upper level is small and of higher order. Hence we have to treat only the two lower right-hand 2 X 2 matrices which are indicated by dotted lines in table 3. (b) We neglect Fr and Fu. The sechr equations can be solved by standard methods. We just give the results. We first introduce .. two quantltles uIz and crti defined by been exact

%z = (4x - A,)/&&

+A>$ + F’I

Tz0)

I -(PP- fg&Jq{z I I

+ ~&&0

WV’%,, &&,A&

W + :x,PnW~~ + ,1, - ~FII ; E ,______-__-_______---_-__---

OI.,= (&

wm4,, - Ayyl -__--_-_-__--

+ F’II

W~NA,,

+ A,) ________

mm4y.y

- Q.)

- +R$“qb

+ F[[

branches: Elr = -1(AFf,,)gil t l&$ t&il”)vG~;,

(74

and similarly for subspace lk: 6’I1 = -#-QJ~~

f t@ - #,~z.

U’b)

For ths coefficients in the wavefunctions, we use the notation c+~ for the coefficient of lore) a(n)= I+) cf(n); C for the coefficient of T,(e) p(n), etc. An ZR upper right and Index t or - indicates the hi$ter and lower energ branch. From the secular equations and (7a) we find for space Iz:

+ g,P,), -B&J.

(6)

They turn out to describe the range within which the wavefunctions change rapidly in character in the LAC regions. For subspace Iz we find for the energy of the two

From @a), (Bb), (9a) and (9b), it follows that in the high-field approximation all the coefficients depend on the two ranges uIz and ollr and on +.h,cdistance to the LAC centre. Moreover the formulae for the two cases k and Ilz are completely analogous.

J.P. Colpa, D. Sielrik/ONPos

278

o corraequerrccof the non-crorsi?rg

In fig. 1a (upper part) we have d:awn fi and E;[ for ff12= -7.07 gauss, uIIz 2 aUTz= -1.77 gauss. One sees clearly that the curves with the smaller u and hence with the smaller LAC range have the larger curvature at the L4C point, connected with a less pronounced repelling of the anticrossing energy levels. We mention already here a relation between the coefficients which will appear to be important for the line shape of the ONP. From (8) and (9) one derives with some straightfonvard algebra:

0.0528

-2Ob

0.947’

32

0.8536

0.1464

0

0.5000

0.5000

+=b +201;

0.1464 0.0528

0.8536 0.9471

nrle. I

The change of one type of wavefunction to the other is rather sudden nfar the LAC point. We give an examplc in table 4.

= .40;&;[z

+ 4-H&) =qI.

(lob)

The half-widrhs of the lorentziansLI and L,, are 21u,_I and 21u,,_ I respectively. They are normalized such’that they h&e the same peak height (I/4), not the same area under the curve. We need further some relations which can be derived easily:

.

0.2 -

;: I t S

@I

(CJ$

= (C,)‘,

(Q2

= (C$

(C;,z = (C:,)? I

(CJ2 = (C?‘,L

(11)

-

(C_‘,j2 + (c;J’

= 1,

(c_;)* + (C$)Z = 1.

3. The field dependence of the ONP in the LAC region and in the high.field approximation

1

I

-10

,

0

,

Mgnetic field -

1

,

I

I

‘f/-e[Go%]

Fig. 1. (a) hchnticrossing in the subspacrs J.z(solid lines) and IL- (dotted lines). The levcl-crosringfield H~c is taken 3s tllCoripin. The valuesof the paramctcrs uc given in section 3. (b) The Contributions to ONP according to cq. (17) from the wwhwion of subspace b (solid line) and of subspacc 1lz (dotted line) Forp, = 0;~ = I;p_ = 0. (c)-h superposition of xc ho &es of fig. 2b.

We use the optical excitation cycle as introduced in [ 1I] and 1121. In the present paper we neglect relaxation effects, they will be introduced later. Eq. (5) in [ 121 now becomes dNJdf = Pi - kiNi, in which P. are the population rates of the excited state sublevels: (six in our case) and ki are the decay rates from these !evels to the ground state. Ni is the population of the sublevel i. In the steady state we have during the excitation cycle dNildr = 0 for each state i and we get Ni = Pi/ki.

(12)

I? [12] we introduced also the decay constants k’ and kb which give the decay rate from state i to a grou:d-

J.P. Colpa, D. Steirlik/ONP

279

as a consequence oJ ~irc rron-crossing rule. I

Table 5 i

--

$

Pi

k’ P

ct

1

a@, +P”)

2

b@, + Py)

0

$(k,

3

ICZp12a@,+Py)+IC;a12fPL

I C,12kZ

K_g?~(k,+Iry)

4

Ic,12$@IF+Py~+IC;p121P~

5 6

IC$l’b@, +Py)+12PzlC;$* ICfp12a@,+Py)+fPilC;~lZ

iC,l’f(k,+kY;y) IC;12f(kX + kv’ I C;Jzx-z

IC,plzk, IC;p12k, IC:,+2f(k, + kY’

0

4(kx+ky,

+ kyl

Par tlrc meaning of the symbols SW the main text. state molecule with nuclear spin cLor fl respectively. It is clear that k,. = “; + k;.

(13)

With these definitions one obtains for the nuclear polarization rate per created triplet state per second [9,11,12] 1

i+(O)=Ru - Rp =

c “~Nii

&pi. i

(14)

RQ and Rp are here the return rates to the ground state with (Yor fl nuclear spin respectively. &(O) is proportional to j(O), the experimental nuclear polarization, which is normalized to the number of proton spins present in the sample. p(O) is also proportional to the light intensity and the quantum efficiency of the crossing to the triplet state. As in [j2] we are going to express the Pi’s, the kk’s and the k’ ‘s in terms of p_ty py, p, or kX, kJ, ki and P. the coeffiuents of the various terms in the wavefunctions. We recall that p,, p , pz are the relative probabilities with which a pure& electronic (hypothetical) TX, Ty ot T_ spin state, respectively, will be populated; p, +p,, + p, = 1. A similar definition holds for the absolute decay rates Q., kY and k,; we have kX + 4, +kZ = k in which .k is the total decay rate. We use also the relative decay rates kk, k’. and “5 defined by k: = C;_Jk, etc. [ 121; one gets then Xr ‘_Yt k; t kI = 1. In order to express all electron spin functions in TX, TY and T, we need the expressions: I+)= loI(

the third of the six wave functions denoted by ti3 in fig. 1:

-(l/&)r,

- T’kPTY

P(n) + CiDTZ o(n).

It follows that K&l’ = K&l’ = jIC_bl’, and IC;.j’ = IC;,I’. The notation used here is introduced in [ 121. With the definitions in [12] we find: p3 zp3 = L/c12 .r y 4 -p ’

Pi = ilCJ2

and

k3=X-2+k3

P’

In table 5 we summarize the results for all p,‘s, kh’s and kb’s. We used the following numbering: i= I,4 and 5 are the wavefunctions of subspace 1:: 2,3 and 6 of subspace 11~. $, = I -Cla(n); 0, = I +) p(n). The other four functions take part in the LAC; the functions 3 and 4 give the lower energy branches (with - index), the functions 5 and 6 give the higher energy branches (t indes). See also fig. I. With these values of kt and kk one finds for (14) after introducing the formulae (11): $0)

= Re - R, = ;(kx + “JN,

- Nz f N41C;D12

- (i/v?)TY, +Ng(c;+J -,lc;J2

- NgIctp12)

(15)

I-)= I/3P)=(1/J2)TX - (ilJz)Ty. We

give one example, for which WP choose (arbitrarily)

t ~~(N3\Cfp\’ +hklC~~12 -‘~~lc_~17- -‘jlc~~l’).

f.P. Colpa, D. Stehlik/OrVP as a consequence

280

In (15) it is only assumed that the high-field approximation is valid but not that relaxation is neglected. We apply (I 5) in the present paper neglecting relaxation. In the next paper (15) will be used for a special case of selective relaxation which enhances ONP. When one substitutes the expressions from table 5 in (13) and (12) and the lvis so cbtained in (15) we get after rearranging terms:

-51 -

Tkf(q.+k;,, fq,(k: t k;, -

f qq.

+ qJ(%,

-

P, -

X’ 5 )

) (16)

PJ

In (16) it is immaterial whether one uses the absolute or the relative decay rates as only ratios are important. The lorentzians L, and LU are defined in eqs. (lOa) and (lob). From the tirst term in brackets it follows that the subspaces Lzand ILz give contributions to ONP which differ in sign. From eqs. (4), (5), (6jand (10) it follows further that these contributions differ in width and in the location of their centres. We specify now as a special case: an unselective decay I;,’ = kb = ki = l/3, but a selective population of the electronic triplet states. Substitution in (16) gives then: &(O)= R, - R, = ($ - $,)(2~~

- P, - PJ

of the notbcrossing rule. I

two LAC centres. With smaller splitting,& the result of fig. lc resembles closely the experimental result (see fig. 2 of part III [7a], bottom,Hllz*) with a positive polarization peak within a broader region of negative ONP. Note that the electronic spin selectivity affects only the sign and amplitude of the ONP but not the shape of the field dependence. Sign inversion is governed by the relative Importance of the terms 2pr and pX + pu. Hence the shape oi the ONP field dependence can be analyzed independent of the electronic spin selection rules. As we have seen it can be characterized by the splitting between the LAC centres HLr& and HF&, (4), and by the ranges uIZ and oUZ, (6), which depend on the elements of the hyperfine tensor only. It is then possible to obtain spectroscopic parameters from the ONP data. Here we would like to emphasize the inTormation about the hyperfine tensor which is available directly from the ONP curve. In general, the ONP field dependence will render a broad (b) and a narrow (n) contribution of opposite sign in the region of electronic level crossing. It is also possible to assign the relative positions of their centres *:tX and HtAc.Taking the example of fig. 2 we obtain: $A,

(lga)

XW.

This inequality constrains the choice of the sign ofA,, and the relative importance ofA,, +A andAX.~ A,,, in the following combinations oi .h$qualities: A _~ L- >o:

H;,,

= H&,

IA,, - A,,1 > IA,,

+ A,,I,

WI

- A,$

(18~)

A,, < 0: HtAC = Hf&

[Note: It is sufficient to take k: = 1/3,.k: t k:, = 2/3, to get (17).] In many casesi, ~0, andp, = I orqv = I, and hence 2p, -p, -p, z-1. Suppose rurther that d,,AY.,, andA,, are ofthe same order ofmagnitude and all negative. One gets then: lolZI 9 JullZI andU& < HpAc. We get then a broad negative Iorentzian centred at HpAC and a narrow positive lorentzian centred at lilck. These results of eq. (17) are plot ted in fig. lb. The parameters used areA,& = -17.5 MHz,,4 ,,//r = -lo15 MHz, A,,/h = -45 MHz, pY = 1, uIz = -!07 gauss, uIIZ = -1.77 gauss. For the purpose of demonstration we choose a rather large distance between the

kx

+ AJ

’ ~4,

This represents the minimum information that can be extracted from ONP data for Hz concerning the signs of hyperfine tensor elements. In (16) and (17) the main approximation is the high-field approximation. We checked the results of (16) and (17) with computer calculations in which the spin hamiltonian was diagonalized exactly (as in ref. [12]). For \EI = hlD1 it was found that&(O) calculated with our perturbation theory had at most a deviation of 1% from the completely accurate calculations in and near the LAC region.

J.P. Co/p, D. Sfcldik/ONP as a consequence of the ml-crorshg rule. 1 4. A more pictorial interpretation of the ONP in the

5. ONP with the polarizing field in the molecular x or

LAC region

y direction

We will try to give a somewhat more intuitive approach towards an understanding of ONP in the LAC region. We refer in this section in oarticular to fig. 1. We have assumed that A,, < 0 so that H& < $& (see eq. (4~)). Suppose we have a polarizing field just at HFAC. We assume that also A,, < 0 and A < 0. From the formulae (6) (8) and (9) it followsiien that at AMIZ= Clone gets:

5.1. htroductory remarks

$s = -2-‘/2[ -) a(n) t 2-li’TZ P(n) = -4T .r a(n) + 4iT o(n) f T-112T3 P(n). J Similarly: $4 = iT, a(n) - ii7” u(n) f 2-“‘T,

P(n).

The coefficients of T=,, Ty and Tz have the same absolute value in $J~ and $,: respectively, and hence both states become equally populated. Furthermore we note that G4 and $5 have the same total probability of l/2 for a(n) and P(n). Hence during the return step of the optical pumping cycle, the states 4 and 5 produce as many (Ynuclear spins aso ones in the ground state and hence create nc nuclear polarization at HEAC. The same is true for 9, = 1t) cc(n) and I$* = ]t) a(n). At HpAr, ONPcan only come trom the remaining levels 3 and’6. At Hr&, outside the LAC region connected with subspace II,, the wave functions are in good approximation: Jr, = 1-j O(n) = 2-“‘T,

P(n) - i2-“‘T,/3(n),

Assuming a selective triplet population via the TXor TV level, e.g.,py = 1 as in fig. 1, level 3 but not level 6 will be populated. Depopulation to the B(n) ground state cannot be compensated by any other decay path and results in an excess P(n) production at HLZAcand hence a negative ONP. With a similar argument one shows that at H& an excess (Ynuclear spin is created because the function $5 = IPP) Ly(n)aWLI&, and this gives an uncompensated (Ynuclear spin contribution when p, = 1. This explains the positive peak at H& and the negative one at HFAfAC.

281

The results of the sections 2 and 3 cannot be trans. formed in a simple way to the ONP with a field in the molecular x ory direction. There are two main differences: (1) At zero field, the energy of T,, is D + E and of T, is D - E when T, is chosen as a reference point for the energy. For a field parallel to they axis, HV, one gets for E > 0 one brach which increases in energy with increasing Hy , beginning at D - Eat zero field and crossing (or anti-crossing) the lriglrer (field-independent) T, level at D + E. For a field in the z direction, H,, one has a branch which decreases in energy beginn& at D - E and which crosses the lower lying TZ level at zero energy. (2) For a field Hy the energy has to increaseby the usually small energy difference 2E whereas for Hz the energy has to decrease by D - E to get crossing. For all cases of interest D - EB 2E, For Hz we showed in section 2 that the electron spin functions in the LC region are approximately hi& field functions. For Hv however they are (neglecting hyperfine interaction) nerther approximately zero field functions nor high field functions at the LC field. For E < 0 similar remarks may be made interchang ing x and y. We sketch in section 5.2 what happens when E > 0 and Hv is the polarizing field (see fig. 1). In section 5.3 we consider the case E < 0 with H, as the polarizing field. These two cases are very similar and hence section 5.3 can be merely an indication of the transforrnation of the one case to the other.

5.2. ONP with polarizing field Hv; E > 0 In the appendix of [12] various transformations are given to spin functions quantized along a magnetic field in a z’ direction which has direction cosines p, 4 and r with respect to a molecular x, y, z frame. We have z’=y and hencep=r=O,q= 1. The transf’onnations in [12] give then: TX*=T,,T$=-Tz, Q’= 2-‘l2 (fy t i/j),

T,.=T$ p’ = 2-“2(ia t fi).

202

J.P. Colp~, D. SrelrliJd0NP as a consequence

l%e IY’and p’ are the proper spin functions for the Z’ = Y axis. The TX,, TY8,and Tq, are defined in the usual way but now in terms OTII and 0’ functions. Analogous to section 2 we introduce a subspace ly with functions TX, a’(n), 7”. a’(n) and T,, D’(n) and a subspace IIy with TX&n), r,,, $(n]and TZ.~‘(n). With these’functions and the transformations mentioned above, the spin hamiltonian (2) leads to the matrices giveri in table 6. In this table we have H, = HZr,A,.,, = =Aby. We shifted the dia= A,,, and AZIZl A,.~~ A.$,,, gonal terms such that the pure TZ, terms get zero energy. WCleft out the tcrmsg#,H,,. These terms are very small and give unnecessary complications in the formalism for a non-high-field approsimation. Note that we go from the first to the second matrix by changing the signs of.4, and A,,. When we neglect the hyperfine terms one gets a level crossing of the T, I = TJ level (energy zero) wilh a branch that begins at energy -2E and lhcn increases in energy with increasing H),. Standard melhods give that the crossing takes place at qc

q

(l/EB)&E)(U

lcE).

(19a)

We dcnolc the two energy branches which bean at TX, and Tig again by u (upper) and Q(lower). (Note that for H,, < Htc the energy of the constant T,, level is higher than the variable energy of the “upper level”). When hyperfine interactions are neglected one gets for H,, = H& for the coefficients C,, of TX, and C’. of TV,:

(19b) + 3E),

C_;,= t/(D f E)/(D

•t 3E).

fable

6 Spin hamillonian

Tar a field in they

direction

for the 1-v and 11~

subspaccs

-2E

-i(-hUr -(D + E)

rule. i

Instead of pure high-field or pure zero-field functions WCare going to use as electronic basis functions the proper eigenfunctionsat the LC point. They are: Ii-’ = C:,T,0’(n)

+ C;.Ty’ a’(n),

1,IY = Ci,T,. a’(n) + C9T , cy’(n),

(204

Y Y

i

T,, O’(n),

and I1lY P = C$T_Y,.P’(n) + C,“#TyOP’(n), Pv u

= C,“,Tx , b’(n) t C” T , B’(n) Y’Y ’

(20b) 1

TZ, a’(n).

We can transform the matrices in table 6 to the new basis defined in (20a) and (20b). When we proceed then as in section 2 and define the LAC fields as the fields at which degeneracy of diagonal elements occurs, we find for the LAC fields in this case; H&

= [-;A Yy + d2.W

+ 01 /gP,

(21a)

and H$

=

@lb).

[+:A,, +~]kP.

We define in analogy to (Sa) and (Sb) 4qr

= HY - H&

(22a)

qi, = i&?.EJ(D + 3E),

Cl,“,= d(l(D + E)/(D f ZE), Cz, = i&E/(D

of the nomuossing

+@Hy)

-!& -:+x 0

MII,, = HY - H$

Wb)

When we transform to the basis (2Oa) and substitute (21a) and (2?a) we obtain the matrix for subspace Iy as given in table 7. A similar spin hamiltonian can be derived for subspace Ily by substituting 4HIrY for AHHrrand -A,, for A,,. The matrices so obtained are formally analogous to the ones in table 3 and from here we could give a treatment analogous to the one given in the sections 2 and 3. There are some important differences compared with the previous Hz case. There, the matrix element which prevents the crossing in subspace I, is (see table 3) (24)- ’ ($,y +A,,). For the present 1~ case the anticrossing is caused by an element &Q.k=-

~A,,~E)/fi

-t 3E.

An analysis which we will not give here in further detail,

I.P. Co/pa, D. SrehliX_/O.VPas a ConseqirerIcc

of Ihc florr~crossing nrle. I

Table 7 Spin hamiltonian for subspace Iy with HfiC as centre of the magnetic field

-.

283

---

0

shows that the difference in the sign of the linear combinations of the tensor elements is due to the fact that in the H, case the anti-crossing is with the lower field dependent level, and in the H,, case with the upper field dependent level. The coefficients of the tensor elements are only equal in magnitude in the high-field approximation, which is usually not valid for a Hv anti. crossing case. We work out the wavefunction near the level-anti-

crossing point in exactly the same way as was done in section 2. We define the range uIY analogous to the definition of alz in eq. (6). We get

Rearranging terms gives:

here are correct only at the crossing field ffcc, (I 9a), assuming D > E they are then far from the high field functions ILU’Q’) and I$$). The level crossing takes place at a relatively low field and the exact electronic spin functions (neglecting hyperfine terms) are still field dependent. With (sly and ull,, as defined here, one describes the ONP only correctly in the region around the LAC pojnts. In the perturbation treatment one may fiid that the ONPis not yet zero at zero field as an exact treatment would give. Usually, however, the ONP is already small at ffu = 0 in a perturbation treatment. One can prove that the field dependence oF the purely electronic spin functions and the behaviour at H,, =0can to a large extent be taken care of by a reoefmed aI,, and CI,~_~ such that these quentities become field dependent. Although some formulae have been worked out along these lines, we do not want to include this rather technical point in the present paper. At or

and similarly

We can handle this set of parameters as in the sections 2 and 3. There is however one important difference. For Hjlz the electronic functions If), I-), and T, are very good approximations

for the electron spin wave-

functions at the Lc point (see section 2) when D 9 E and these functions are field independent. The functions f’y ’ u Introduced P ’ fry u ’ f”y, P and f”y

near the LAC points the situation is very similar to the Hz case and for numerical details the general model 1121 should be used. In the sections 3 and 4 we have shown that at H& a negative polarization occurs when TX and/or 1;, are active in the intersystem crossing (p, + pY = I) and the sign of the polarization is reversed for the opposite selectivity when T, is the active component @= = 1). At HIIZ uc one gets a polarization opposite in sign to the one at If&. Similarly we get a positive polarization at HP& when p = 1 and a negative polarization when P, + P; = 1. A:H$ one gets again a con tribution to the ONP with the sign opposite to the effect at HLC.

J.P. Colpa, D. SrehIik/ONP as a consequence of rile norwrossing mle. I

2&l

For a field 1!, the transformations in [ 121 mentioned in section 5.2 get direction cosines p = 1, 4 = 0, r = 0 and hence o” = 2-19,

p” = 2-t/+,

$ p)

f 0)

T,u = T,.

Ty,, = Ty,

TX,.= -Tz,

like H& and aTX,etc., are independent from the rate equation (15) and they are still good parameters for the H, or H cases. In particular it is possible to take over the SCK eme of inequalities given in eqs. (19) forHllz in order to obtain information about the sign of the hyperfine tensor elements. We consider the situation of an ONP field dependence with a broad ONP contribution upfield and a narrow contribution downfield quantities

5.3. ONPwith polarizhg field H,; E < 0

There are spin functions defined with respect to a Z” usis which

coincides

with

the molecular

(26a)

x axis. When

one

works out the hamiltoninn along the lines given in section 5.2 it appears that the following substitutions have to be made in the formulae forHY, 6 > 0, in order togettheonesforH,,E
Ayy>O: + doubly dashed

functions, e.g.,

TV8+ T.yro, etc., 2E+D-E,

D tE+-(2E),

A -)A.w AZ +Ayy, YJ This leads to: Hfc = d-lE(D

=

H$=H$,

IA,, +AxxlX4A,Z -A,I, (26b)

(24) A,, +A,=.

- E)/gj,

(25a)

ollx

Using (21) and (23) we conclude to the following set of incqialities:

&A,,,,= - Azzd!~).(Xb)

All details are very much the same as discussed in section 5.2. However, we mention one aspect in particular. For H, and E < 0 one gets a positive polarization at H&when p, = I and a negative polarization when P.+P; z 1. The contribution to the polarization at Ht& has always the sign opposite to the one at HbC as before. Formula (15) and hence (16) and (17) are derived for the high-field approximation, which is usually correct forHr. Formulae like (lS)-(17) for low-field cases (H, or HY usually) are more complicated. Eqs. (16) and (17) however still give qualitatively the correct behaviour when applied to the low-field LAC cases. The

It is easily recognized that certain combinations of the inequalities (18) and (26) are ccntradictory. Assuming, for instance,A,, >O, (18b) requires a different sign for A,, and A . Hence for A >Oweconyr e eq.(26b) reqzres the same elude that Axx < 0, whi sign for A,, and A,,. Similarly,the other combina. tions of signs of the hyperfine tensor elements are mutually exclusive. Hence we have to conclude that the assignments (Isa) and (26a) cannot be correct simultaneously but that one of these inequalities has to be inversed. We would like to emphasize that the second level crossing situation can be analyzed quite analogously although the hi&z-field approximation is not applicable at all. Note also that the particular conclusions given above assume a hypert’ine tensor with a principal axes system coinciding with that of the zero-field-splitting tensor. 5.4. lie nowcrossing cases Hy arid E < 0 or Hx and E>O For these cases no simple perturbation treatment is available and we calculated the polarization using the general model given in ref. [ 121.

J.P. Colpa, D. StchlikJONPas a cot~scquenceof rhc nomcrossingrule. I

6. Numerical examples of ONP as a function of the polarizing field 6.1. Anisorropic hypcrfine temor

As mentioned before, one can diagonalize the spin hamiltonian exactly and consequently solve the rate equations [12]. In fig. 2 one finds the results for a particular case. We took the following parameter values: D/Jr = 394.8 MHz, E/h = -15.3 MHz, A,$ = -7 MHz, Avv/h = -21 MHz, A,#i = -14 MHz,@/h = 2.8 MHz/gauss. We further usedki = F$,= ki = l/3, and px = 0.05,~” = 0.9,pz = 0.05. The zero field parameters have been chosen to rem produce the level-crossing regions described in the experimental part III [7a]. Somewhat arbitrarily the

2B5

hyperfine tensor has been assumed to be that of an aromatic C-H fragment. The chosen order of magnitude of the tensor elements has to be at least as large in order to explain the experimental results. However other combinations of the relative size and sign of the tensor elements are possible too, as will be discussed later. The electronic selection rules are chosen to rcproduce the correct signs of the ONP. The numerical results apply also to the case with the sign of E inverscd if we carry out the following substitution: .Y*JJ. With E < 0 we get LAC for Hz and Hv. With the formulae (3), (4) (5) (G), (16!, (17) and (25) and the parameters given above we fmd the following data: H’ = 141 gauss H” = 143.5 gaus~,H~‘~ LAC = 138.5 ,a& UIZ= -7.& 1;ucss; UIIz = -3.536 gauss. We further find for H_:

(27)

Fig. 2. Calculated ONP tatcsbT(O) per created triplet state as a h-tction of the polarizingfield H oriented along the three

principaldireclions.All diagonal&nents of the hyperfinc tensor are tkan to bc different. See section 6.1 for the parameter vducs.

Formula (27) represents accurately the exact computer results for H,in fig. 2. With (2G) one finds a minimum at 128.5 gauss (&(O) = -15.0 X 10m3/s)a maximum at 138 gauss (‘j,(O) = 75.9 X lO-3/s) followed by another minimum at 145 gauss (j+(O)= -154.8 X 10-‘/s). One compare this with the lower part of fig. 2 which gives the computer results. We further calculate: Htc = 40 gauss, Hr*, = 41.25 gauss, HL& = 38.75 gauss, uLr = 5.60 gmss, and uIIx = 13.37 gauss. As mentioned at the end of section 5.3 one expects at HF& a negative polarization when p dominates the selectivity. It means that we espect a zinimum around 41 gauss and (see section 5.3) a maximum around 39 gauss. In our case we have lolls I > lokland hence one expects the positive part to be broad, and the negative contribution to be relatively narrow. This explains the behaviour of the calcuIated ONP curve for H, in fig. 2. The ffY case is not caused by LAC when E < 0: and the corresponding curve in fig. 2 is computer calculated. Note that for Hy the magnitude is almost two orders of magnitude smaller than for the other two directions with LAC.

J.P. Co/pa, D. Srehlik/ONP a5 a consequence

286

olrhe

non-crossing rule. I

6.2. ONP with atzisotropic hyperfine tensor

Isotropic or nearly isotropic hyperfine tensors may occur in some triplet states. In liquids the isotropic part of the hyperfme tensor will be the more important part to consider and as it has been suggested [ 161 that ONP has been observed in solutions it seems appropriate to make a few remarks about this special case. When A,, = A,, = AZ, it follows that ulli as defined in (6) becomes zero, and hence, one would expect no LAC contribution to ONP at iYF&. From (23a), (23b), (25d) and (25e) one sees that neither of the other u’s will be zero in general. We give an example in fig. 3. We have chosen A,,fh = A,_Jh =AJh = -14 MHz. All other parameters are as in the example of section 6.1. Our formulae give in this case: H;ic

= 143.5 gauss,

Hkc

f: (0)

= 138.5 gauss,

52 = -7.07 1 gauss,

Qz = 0.0 gauss,

HpAc = 12.5 gauss,

H;$

Ulx = 6.9 gauss,

%v = 12.1 gauss.

= 37.50 gauss,

For H, the pammeters are of the same order of magnitude as for the anisotropic case treated in section 6.1. Hence it is not surprising to get similar ONP curves for the two examples both with positive and negative contributions. The lack of anisotropy is more evident in the Hz case. With u,IZ = 0, our formula (I 7) gives with the relevant parameters:

Fig. 3. Calculated ONPrates frT(0) per created triplet state as a function of the polarizing field H for the two crossing dircctions. The hypcrhc tensor is takcfto be isotropic. See section 6.3 for the parametervalues. functions become identical only when E = 0. We could have transformed the matrices used in section 2 using lu) and lg) instead of I+) and I-). When we neglect ,Q,HZ we would have obtained the following results by using lu) and 111)as electron-spin basis function:

This describes in fact the broad negative ONP in fig. 3 (Hz-case). One finds however with a very accurate com-

puter calculation a very narrow positive peak at 138.5 gauss although uIIi = 0, according to (6) when A,, = A

. ‘fhe

origin of this peak can also be understood from a perturbation treatment. At the LAC point (field Hz) the best electron-spin wavefunctions are C,“T, + C,T, = Iu ),

C;T,

+ C;Ty = 111).

(28)

in which Cl, Cy, Ci and C,’ are defined in the eq. (3b). The functions / +) and I -) used in section 2 are good approximations for I u) and Ii?)when E
and HIrz = HZ - (&,

+ =>I&

One would have obtained for the subspace IIz the matrix in table 8. This matrix is similar to the ones in table 3. For subspace Iz one gets a similar matrix, to be obtained from table 8 by changing the signs ofA,, and *,,.

J.P. Colpa. D. StchlikfONPas

CTmnequencc

of the non-crossing rule. I

287

Table 8 Spin hamiltonian for subspace Ilz but with improved wavefunctions as a basis. qj = Hjj

lk

IQ)P(n)

lu)P(n)

I I

Proceeding as in section 2 one finds for aIIE a definition differing slightly from the one given before in (6).

Usually E Q D and one makes no serious error to use uIIz in definition (6) instead of (29). Wflen A,, =Ayy however one gets

When Ef 0: uiIZ i 0, although usually small. It means however that one gets LAC at IY& also when A,, = Ayy. With the parameters used in our example one finds uiIZ = -0.14 gauss so that the narrow peak in fig. 3 has a half-width of = 0.3 gauss. It is ciear from this that in case the hyperfine tensor is isotropic, one may find extremely narrow peaks in the ONP. When A,, #A,,, the refinement described here is numerically of no importance.

7. Discussion and conclusion The main purpose of this paper is to establish part of the conceptual framework within which the very strong ONP effects can be understood in relatively simple physical terms. L4C due to hyperfine interaction is clearly one of the causes of large ONP. Although numerical results can be obtained by computer calcu-

Tz 44

0

lation, this method has the disadvantage 0: the black box character whereas perturbation theory gives an adequate method to defme important physical concepts such as the level crossing and level anti crossing fields and the various ranges as the most important parameters for the description of strong ONP. The field dependence of ONP, sometimes changing sign a few times in a very small field region, which after the discovery seemed so erratic, has now a natural enplanation. It is the superposition of two lorentzian type functions differing in width, in sign and in centre, all as a consequence of LAC. Hence we conclude that the typical ONP behaviour is reflected in the results of the LAC theory presented here. It should be mentioned that ONP without LAC is certainiy possible [12]. The strongest ONP so far observed is however related to LAC. WMe for Hllz the experimental (fig. 2 of part III, [7a]) and theoretical curve (fig. 2) agree in quite some detail, the agreement is poor for lhe second levelcrossing case. Furthermore a significant difference in the absolute polarization rate between the crossing and the non-crossing case is not found in the experimental results. Before we can discuss these features more explicitly, we have to treat first the crucial role of relaxation in the shape and strength of the ONP effect. This till be the subject of the next paper. Other aspects have to be considered such as the influence of non-diagonal terms in the h-f. tensor and the effect of more than one nucleus coupled to the triplet. Preliminary studies show that these aspects may have some influence on shape and intensity, they do not change the essentialfeatures of the theory givenin this paper.

188

J.P. Colpa. D. Stehlik/ONP as a consequence

References

111(a) G.

hiaicr, U. Hxberlcn, H.C. W’olr and K.H. Hausscr, Phys. Letters 25a (1967) 384. (b) G. hlaicr and H.C. Wolf, Z. Naturfschg. 232 (1968) 1068. (a) tl. Schuch, D. Stehlik and K.H. Hausscr, 2. Naturfscbg. 26a (1970) 1944. (b) P. bu, D. Stehlik and K.H. H~~usser,J. Magnetic Rcson. 15 (1974) 270. I31 A.H. hlaki and J.U. van Schlitz, Chc.11. Pbys. Letters 11 (1971) 93. I41 P. Lau, Docroral Tbcsis, Hcidclbcrg University (1975). 151 P. R&h, Diplam Thesis, Heidelberg University (1975). 161 G. Dillrich, Doctoral Thesis, Hcidelbeq University (1975). (71 (a) D. Stchlik. P. R&ch. P. Lau. H. Zimmermann and K.H. Hausscr, Cbcm. Phys. 21 (1977) 301. (b) G. Dittrich, D. Stchlik and K.H. Hausser, to bc pub lished. Ml K.H. Hausscr and H.C. Wolf, Advan. hlagn. Reson. 8 (1976) 85.

of the non-crossing rule. I

[9] (a) D. Stehlik, Habilitations Thesis, Heidelberg Universiry (1975) also published as: (b) D. Stehlik, The Mechanism of Optical Nuclear Polarization, in: Excited Stales. Vol. 3. cd. E.C. Lim (Academic Press, 1977). [lOI J.P. Colpa, K.H. Hausser and D. Stcblik, Z. NalurForschung 26a (1971) 1792. [ 1I] J.P. Colpa and D. Stchlik, Z. Naturfoschung 27a (1972) 1695. [ 121 D. Steblik, A. DochrinS, J.P. Colpa, E. C~llapbw and S. Kesmarky, Cbcm. Pbys. 7 (1975) 165. [ 131 W.S. Veeman, Tbcsis, University of Leiden (1972); W.S. Vceman, A.L.J. van der Poe1 and J.H. van der Weals, Mol. Pbys. 29 (1975) 225. [14] R. Furrcr, J. Gromer, A. Kachcr, hl. Schwoerer 2nd H.C. Wolf, Cbenl. Pbys. 9 (1975) 445. [ 151 A. Carrington and A.D. hlclachlan, Introduction to hlagnetic Resonance (Harper and Row, New York, 1967). [ 16) J. Bnrgon and KG Seifert, Bcr. Bunsenpes. Pbys. Chem. 78 (1974) 1180.