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Theory of modulation effects in electron electron double resonance
CHEhiICAL PHYSICS.LE’ITERS :
Volume 28, number 2
I.5 September
1974
THEORY OF MODULATION EFFECTS IN ELEC’RON ELECTRON DOiJBLE RESONANCE Bruce H. POBINkN; Departmkt
iean-Louis of Ciiemirtry,
MtiNGE*;
Luraine
Vanderbilt University.
A. tiA&TON,Larry
R. DALTON
T*
ffaslklle, Tennessee 37235, USA
and Alvin L. KIVIRAMi Department
of C?lemistry, University of Wcshington, Seattle, Washington 98195, US4
Received 7 June 1974 The nonlinear revonse of spinsystems to intense radiation fields is quantitatively treated by a modification of the stochastic Liouv-Ue equation for the spin density matrix. In particular, applied modulation terms are included in this equation. The resulting formal2.m provides a general method for calculating nonlinear spin response for dilute systems of idiuls in a high magnetic field. In this communication, frequency and field’swept absorption and dispersion electron-electron double resonance spectra are ukulate$ and compared with experimenta: spectra recorded under conditions of sinusoidal magnetic field modulation and phase-sensitive detection. Good reproduction of the detailed line-shapes of experimental spectra is observed in all casee. The dependence of ELDOR reduction factors upon modulation frequency is discussed. A theoretical analysis such as e,nployed in the present cJmmuni&tion is shown to be essential i.fELDOR reduction factors are to be,related to relaxation times and hence to molecular dynamics, and if the design of ELDOR experiments is to be optirmiied.
1: Introdl,ction
The response of spin systems to strong radiation fields has been shown to.be dependent’upon (1) spinlattice relaxation processes, (2) spectral diffusion processes which transfer energy throughout the reso; nzrce spectrum’(e.g., between resonant and nonresonant spin transitions), and (3) applied-magnetic field ., modulation of the resonance.condition 11; 21. Further-. more, it has been demonstrated [2] that nohear, methods can monitor sp:e&l diffusion processes for correlation times]significa#ly .long@rthan t&se for ‘. which i&h techniques are useful. In-particular, car-;:’ relation times over’s farige of four brders of nikgnitude centered
at the electron
spin-lattice-relaxation time using nonlinear ._ I_ :
(= 10y5 sj have been imxstigated .-
ESR as well as electron-eiedron daubk resonance (ELDOR) methods to study stable free radicals (spin labels) at low.temperatures [2J. For slowly tumbling riitroxide spin labels (molecular reorientation frequenties less than the frequency corresponding to tie magnetic,ariisotropy of 2rr X LO8Hz) ‘Lhedominant mechanism for spectrel diffusion is modulationof the anisotropic.electrori Zeemen and E4N hyperfime interktions caused by mole&r reorientation (tumblirig). For rapidly tumbling spin labels spectraI diffu-.
sion is determined by spin-flip processes -either nuclear spm-lattice, c+; processes or mtermolecular electron Heisenberg exchange, ; o,;;‘processes. For both the slowly and rapidly tuinblingregimes the applied magnetic field modula: .intramoiecdar
tjon p!aysa crucjal role in determining the character ‘-_ -.of the~nonliriear spin response. .’ : We l-&e &ce
.. ,
V&me 2,8,‘number 2 : .. ” _C,+;\IlcAL PHYSICS_-LETTERS “,_: -. ‘,_i 15 September 197.4 .’ .o..: ‘..,.. .‘.,. : :. : EUOR $nd E&response pfa spin system in the 1 ,- :,,. .’ ‘, ’ +j.= d; [s, e-?Jof+,s._ e’Wor] :sl& tti~ling reg,ir&:In t’lis commuqication we.+ .: p!;. a similr&formali&r to &cu!ate the r.espbnse for -iL+r,+$_ e&p’] + ds~z~[eiW sr+e-iwst] tdp[S+e .ra$dly?uqbling spin labels. For purposes o$ brevity j ‘. -an&ciarity we shall &strict our diskksion to simple ‘. _ : .mo’dels which illustrate most of the results of mole .’ + C iiiKizy [eiiuSr+e-iwsr] ,, ..(3) I( realistic (and more complex) model calcu!!ations : which we sh,all repoit :&-Idetai! eIsewhere ]4]f:The methods.outlined below provide ti powen% tool for _. where& =$yekti,,dp =fy~h,,d~=&,H,,and -the study of the correlation times associated with ’ $r,.,,H,. The frequencies of the’.microwave ‘IL =’ molecclar’diffusion, which gives rise to Heisenberi. .’ observer, microwave pump, and tieId modulation spin exchange or modulation of anisotropic interare designated by w,, , upI and os, respectively. The actions. fist tlwo terms on the right-hand,side of es. (3) describe the interaction ofthe electron spins with the ,’ microwave observer and pump fields; respectively. ‘. 2.,i;enek theory The last two terms describe the interaction of’the electron and nuclear spins with the magneticfield .To analyze the nunlinear response.of a spin system, modu:niion. The co,mmutator of the hamiltonian. f-lR to applied radiation and &odu!ation fields, we em-. which describes the coupling of the spin system to -pi.oy the general Boltznann equation for.the spin ,the la1 ticeis approximated by [7] density matrixas developed by, B!och [S J, Redfield -i[HR,o] =.-r~(u’u’)=-rRX ~ (4) [6j, Abragti [7], and others [8] and modSed tq treat the effects of markoffian rndtion by K&o 191 where go is the eqiilibrium gin density matrix. We and’Freed [lpJ:- . . express the’-off-diagonal matrix elements of this commutabzrr in terms of orientationahy invariant spin&, r) = -i[H(ti,tj,o(SI,t)] s-pin relaxation times and the diagonal elements are ex(1) pressed in terms of orientationally invariant spin:- :r,C& r)- uO(sr,t); ~ ,. lattice re!axation times. where I2 d&otes molecular orientation and )?a is a ,” The steady-state solution to (l), appropriate for time;independeni Markoff npemtor. Invoking the high frequency microwave fields, is obtained by using high field approximation (i.e,, neglecting terms : a series’expansion for X:. S,I,, etc. [l I]).the spin system can be described : by a syin hamiltonian of the forni H(R,
t) = ff,
t: H#k)
:+ c(r)
+, HR
:,
y,(ti,r)y (2)
= CZJQ,w7
a rw,)ei(~7*‘w~r
7
where the time-independent hamilt~oman H,, , which = p F”m
Volume 28, number 2
-CHEtiCAL
of ELDOR signals upon modulation;‘electron spinlattice relaxation, and upqn m&hanisms which transfe.r saturation between the pumped and’observed transitions, let us discuss the simplest system to .which ELGOR can be applied: a four-level system involving two electron spin transitions. We consider a particle described by a time-independent hamiltonian Ho = oPa S z J‘umpinop between two sites denoted by j= 1 or 2 and k= 2 or 1 . Fbr the two-site jump model we have, considering only secular terms, H,Q)=FG)S, where F(1) = -F(2) = F. F is one-half tlie total magnetic anisotropy or frequency difference of the magnetic interactions in the two sites. The operator in eq. (1) whichdescribes the markofsan mction is
r=
-w,
.(
WJ
y-w;
WJ ) ’
15 Se&tber
PHYSiCS LEFERS
+Jiw,Y,(i,Frw,) = dp {Z,,,(k,
1971
(74
wp +-rwJ - ZJ”,
--wpfrw,)
5,
where A, = w. - wpa and q = fi/kT. We have emFlayed the additional definition that the diference between the diagonal elements of the reduced density matrix is given by,
(6)
where oJ denotes the mean jump rate. Given the above expressions for f-lo, HI, and I’ and denoting the two electron spin states as usual by a and B we express (1) in terms of the matrix elements Zga using (5). This ;esults in the iollowing set of general coupled equatioris for the pumped and observed transitions:
Information regarding the magnitude of the respoase. of the spin system resides in the (complex) matrix elements ZpO and we must solve the simultaneous equations in (7) for these matrix elements. However, the Zpcl are not themselves the experimental observables. Indeed, the design of the microwave detectioncircuit together with the use of a phase-&nsitive detector results in the selection of a particular Fourier amponent of the spin-system rer,ponse given implicoutitly .by Z,, e iwf, In general terms, the utitered put of the phase-sensitive detector is given by’
+~~~(~,~rw,)e-~(~~““~)‘~
sin worcos(rws~+
‘.
(9
where the first factor describes the coherent response of-the spin system to the applied time-dependent‘ field and the second factor represents the reference ‘. sgml employed to select a given Fourier component of the resFon=. Thus the microwave sykem is adjusted TOselect either the dispersion signal (coefficierit of the cos uof term).or the absorption signal
=$This four-level model al_~ descibes intermolecular Heisenberg’spin.exchange between two’electr6n sini : (wex processes) as yell as the,gLDOR‘ie@nse for a. coilpled electron-nuclear S=I= l/2 -fin sys,tem domk ‘kted by intramolecular nticlw spin relaxation (wn
procqes).
@I] ) dr;
) ‘.
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Volume 28, number 2’
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(coefficient ofthe’;in w,t tekj, Then rhe phasesensitive detector is adjusted to select the in-phase
such as, for example,
Z(1) *Z(2)
and r(?)
= Y(1) + Y(2).
4. Results and discussion .’
Proceeding from (7) we have numericalJy computed the various l%DOR signals [(combinations of Zpao(+, o_-Ot rwS) and Z&(+, w. - rws)] at the first and second harmonics of w, for coupling to barmom&as high as f = 40 under the condition that the pur+observer’frequency separation is equal to the exact frequency separation between th.e two resonance lines, i.e., I-up - woI = 2F in the case of field-swept spectra and under the condition that w,, = wBu + F in the case’of’frequency-swept ELDOR spectra. The -calculated line shapes.for the latter case are presented in’fig. 1. The agreement with’experimentally observed line shapes is excellent. Likewise, excel!ent agreement is obtained for field-swept ELDOR spectra+: In addition to-the Xne shape information we’also obtain explicitly the’dependence of the ELDOR signal amplitude,.ususlly~expressed in terms of the re-. duction factor, on instrumental and molecular variables. The: ELDOR reduction factor is defined as R = I 7 Izp~o(~;
= ‘WZ9(ro(dp
= 0))
Fig. 1. Computed frequency-swept ELDOR spectra at the first axd second harmonics of the modulation and in phase with tte modulation a.re shown. (A) First harmonic absorption; (13) fast harmonic dispersion; (C) second harmonic absorp-ion; (D) second liarmonic dispersion. Parameters employed in the calculation include (w -‘wo)/Wo = +F/do =’ 2000; 42d, =0.63;d&io= 2;,WJ 72d, - l.l4;4d&T,Tz = 0.062; Tl = 2Tz ;,dp/do = 5; w. = WB - F. Spectra shown represent plot of signzrl (abitrary units) versus frequency separation of pump and observer, with frequency increasing to-the right. ELIJOR Lines are centered at wp = w& + F.
where Zpoo(dp = O), the ESR signal pump off, represents a combination such as ~z;~o(:+. G.‘o+ rw,> + z;,,
the in+ase dispersion signal. Many variables such as .h,, h,: T,, T2, uJ, w,, etc., influence the reduction factor: we limit the discussion to the more commonly encountered experimental situations. Representative behavior of calcuiated ELDOR reduction factors is presented mfig. 2. For low modulation amplitudes the effect of modulation frequency on the in-phase dispersion signal is insigni5cant: However, the-commonly measured absorption signals gre ser&tive to modulation frequency: the ELDOR reduction factor decreases
,’
..
(+, 00 - rw,> 1 2
chA&‘iriW~
increas?
k&w,
T1 (cf. fig--Z),
t& fmcdond char& ii i.nipp+nt even for mJ T1 4 ) It can be shown that thk mlcuiated lineshapes &equally ‘. 1. Nofe too that the reduction’faciors for frequencyvalid Pol.dilute free radicals in the soid state. Therefore, : swept absorptionBLDOR can be significantly less these rcsuits (;1n be used for th6 analysis of ELDOR spe$Fa 1. of randdmly oriented radic&s in the solid phase:. &~n’~th: reduction factors for field&ept absorption ‘. “.’ ‘, .; ‘. IT’, -, .1 : : .’ /, ‘> .. ‘-, ,: .,,‘. ..::..., :’ : j’. .. : : _’ ,, ;,:; _, _,., ,_... ” : I . . . . -. ., ‘.: ...I ... : -:. -.. : _:- : .,,‘.’ ..‘.,; ,. .I .,.. ., ,. .. ,‘. 1:. .:.’ ‘. .:.: ..‘. _: ,‘, : ..,. .” : .“_ -:’
0.05
15 September
CHEMICAL PHYSICS LETTERS
Volume 28, number 2
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UJTl
Fig. 2. The dependence of the ELDGK redukn factor for various in-phase signals upon WJ T1. The open snd solid circles represent the reduction factor for the field-swept (up- w. = 2E) absorption ELDOR signal at the first 11xmanic of the modulation for a value of wsTl .= 0 and for ws5”l = 3, respectively. The dashed line (solid t&gles) IO presents the field-swept dispersion ELDOR signals at the fLrst harmonic of the modulation for these modu!ation frquecties. The open squares represent the frequency-wept absorption ELDOR signal at the first harmonic of the modulation with the observer positioned at w. = apn - F and with w,TI = 0. Othei parmeters utilized in the c&ulation include; F/2d, = 1000,4cI~ TITz = 0.07, Tl = 10Tz; and dp/do = 8.
ELDOR, as is experimentally observed. Also in agreement with theory, we have observed that dispersion ELDOR reduction factors measured for field-swept and frequency-swept &odes are identical. The decrease in reduction factors at high WJ TI arises from the spectral diffusion process shortening the effective T1 and T2, making it more difficult to saturate the observed transition; This effect can be visualized by noting that.in the limit of no modulation we can write the saturation factor as
where n = 2, n! = 1 for the two-jump modei and 12= 3, m = 2 for a thee-jump model (or equivalently for Heisenberg spin exchange coupling the three electron ridiwl). spin transitidns of a nitrotide A point of considerable practical significance evident from the coniparison of reduction factor’s fbr various operational modes is thar’ dispersion ELDOI? &fords significant advantages relative to absorption
1974
ELDOR r&t only.in the ineasurement of energy -.r transfer rates (wJ , w,, c+,) but also in yieIding better signal to noise in the measurement of spectroscopic parameters (hyperftie intera6tiot-G). For this litter purpose, ELDOR is usually performed in *-hefrequency-swept mode by positidnitig the observer at exact resonance for some Gansition. Our theoretical and experimentaYresuIts confirm that better sensitivity is obtained in dispersion mode for rapidly tumbling radicals (and for radicals in single crys!als). The dependence ofR upon $/k;is shown in fig. j for two values of ho. For values cf It, in the linear spin response region (fig. 3A) the general slzope of the reduction factor curves is independent of fl, with the curYes merely being displaced aIong the $,jtl, axis for differen t values of ho. AIso if the saturation factors [see eq. (1 l)] for two- and three-jump models are adjusted to be equal, the calculated curves of R versus ~zJ~, for these two modeIs are virtually superimpo-able. The dependence oFR uporr h,lh, is quite insensitive to modulation amplitude as long as the modulation amplitude is substantially less than the spin packet linewidth; however, as the modr-dation amplitude approaches the linewidth, significant effects are observed. By comparing a series offigures such as those shown in fip. 3 it is readily seen that the greatest effect of modulation frequency on the ELDOR reduction factor occurs at precisely those conditions under which most ELDOR measurements will be made, i.e., near the linear response region on the observer sat&ation curve and for pump-observer ratios between 2 and 20. These results clearly demonstrate that modulation effects play a crucial role in determining R acd cannot be ingored (as has been past practice) in relat,ing ELDOR reduction factors to molecular energy transfer rates.(in .&is case to ws T1 ). It should be emphasize2 that by appropriate replacement of oJ by w, @Wex our calcuiations aIso reproduce the effect of intramolecu!ar nuclear spinlattice relaxation or intermolecular Heisenberg spin exchange for a four-level system. Extension of the calcl$ations to *sterns with mbre than four levels is st+i&tfo&=rd. more
Furthermore,
genera1 spectral
Hfusion
the intrcduction mod&k
readily
of ac-
cornmodated by the fo,n@lism discussed herein. To illustrate this we have &ied out extensive calculations for a:typica:six-level system (S =. l/2,:1 = 1)
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CHEMICAL PPFiSkS
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,. Fig. 1. The
Qlcuhted variation of the ELDOR reduction faGfor_.. .for field-sa.ept absorption
and dispersion ELDOR at the fust harmonic OKthe modulation and. in ph.ax with the modulation versus the rati) $/ho (or, equivalently, +/do).. The circles and ‘quarkrepresent ihe ibso&ionR’s calcr~lated for ws’i’r = 0, and for wsTr = 3, respectively. The dashed line represents the disper_iion reduction faetors;,which axe apprdximetcly independent of &,Tr _(A) is cimputed for a value of ho in the hncar spin response region (no satluatinn) whi!e (B) is computed for’xwlue of ho in the nonlinear region (hard saturation) of the observer mturatidn cuNe. Other parameters used in the c&&ion are T1 = Ta, WJT, = 2.5, F/ado _ = 1000.
:
appropriate for a nitroxide radical. These calculations take int s account both w, and sex spin-flip proces‘ses. Moreover, the two-site jump model has been replaced
by art isotropic
brownian
diffusion
model
in
order tc reflect more realistically the character of the motion which modulates the anisotropic magnetic inteMctions [12]. In fig. 4 we present the theoretical simulation (crosses) of a typical frequency-ssvept ELDCR spectrum for a nitroxide radical in a low .viscolity solvent. The calculated value for the product of the molecular relaxation paramet.ers is We,T1 s 3, in reasonable agreement with the value expected theoretically for this system at T = -22~5°C. In order to illustrate an alternate means of.deterI I I mining the molecular relaxation parameters we re20 40 60 ‘80 produce in fig. 5 representative ELDCR reduction -. fYp-90) in MHz . ., factors measured at modulation frequencies of 270 Fig. 4. The experilyental in-phase disp,zrsion ESR (A).at the Hz and IO0 H-Iz for absorption and dispersion detecfirst harmonic of the modulation’and the in-phase dispersion tion. The theoretically-predicted diminution of ab.’ELDOR (B) at the first harmonic recorded in a frequency sorption reduction factors with increasing modulation srieep mode with the microwave observer at the exact re&-: frequency is obs’erved. When a least squares fit of the nanceofthcm~=-1 r4N hyperfine Line. The sample is a experimental data to theoretical curves is carried out, 3 X 10-a M solution &2,2,6,6-tetramethyl+piperidinol-l.oxyI (TANOL) in set-tutylbenzene .at -22.S°C..The follow.all sets cf data extrapolate to the same R(dp +.-) ing conditions apply: So = O.OOz?rad Hz. ws = 6.28 X IO! value, as.is theqretically required. This value is 0.8, m-d Hz, dp (=0 for ESR. = 5do for ELDOR). The fist index’ ii:hichyields ,for the product of molecular relaxatipn, in the parentheses associated with R indicates the nuclear with the :$Qn quantum.number of the-14N hyperfme hne pumped ‘1 ,. paran’efsr? we.x.Tl = 4; ag~~:cor&t&t ,while ihgsecon3 indicates the.line observed, The crosgs ‘., ,valuis e::pected:for- this system ,at T = Sa,Ci The two r@r&nt theoretk+hy c&r&ted vahks and the’best fi,t car- ,:’ methods illustrated in figs.4 and 5 provide in impor- ‘. -iespo+dsto &.&Fl G.3:. .:’ : ., :‘,‘, .. ~ : -. .. :. ‘j$4”‘_,:’ ..I +._ I
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Volume 28,‘number
C~MI&L
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PI?YSICS LETTERS
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1974
A.&no@edger~ent is’made to the Donors of The Petroleum Fesearch Fund, administered by the, Aiierican Cl?p_mi&al Society, tj the Research Cosporation, and tq the National Science- FoL:ndation @P-28183X arid GP42998XJ for p.zrM support of t?b research. We acknowiedge suppoti for the construction of nonlinear spti response instrumentation from the Cherr&al Inatrumentztion Section, National Science Foundation (GP-3.6774).
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Fig. 5. The variation of experimental field-swept in-phase absorption and dispersion ELDOR redwction fact&s; R(O,+l), detected at the first harmonic of the rno~u~tion versus +/ho (or, quivzlently, +/do). The absorption reduction factors are shown for twd moduhtion frequencies; name@ ws = 6.3 x 10’ rad Hz (circ!es) and ws = 1.7 x f03. rad Hz (squares). The dispersion ELDOR reduction factors, denoted by triangles, were recorded at a modulation frquency of 100 kHz.‘The s-ample is a 3 X 10e3 M solution of TANOL in secbutylbenzene at 8°C. The.solid and dashed lines z& best fit theoretical values calculated for So 4 0.005 and wexTl = 4. The foLIowing magnetic parameters were also employed iE the calc&tion: gxx = 2.0094, gjlv = 2.0961, gzz = 2.0321,g = 2.00587.d,(‘4N) = 18.248 M.Hz.A~~(~“N) = 18.810 hfij~,A~~(‘~N) = 92.646 MH~,F(‘~N)_=.43.235, hil-l~!~Si(~H) = i.35 MHz.
taut test of the theory and in all cases thk results have been consisterit. It is important to note that the ELDOR reduction factor plots contain no adjustable parameters. In order to niake meaningful comparisons of experimentaland theoretical reduction factors one must have precise values for the ampIitudes and phases of all the radiation and modulation fields over the s&nple volume. We have expended considerable engineering effort in cqntrolling and pieeisely measuring these factors. Full details of this work, together with a description of our ELDOR spectrometer, which pas.- sesses ~pab~ities for absorpt,ion and dispersion ELDOR measurements at .the first and secqnd ha& mimics of the moduIation and tit twer-$y’discrete’ modulation frequencies ranging from 35 Hz to ! MHz, will. be presented elsewhere 1131.
[1] L.R. &tori, A.L.Kwiram and J.A. Cowen: Chem. Phys. Letters 17 (1974) 495. [2] J.SI Hyde and L. Dalton, Chem. Phys. Letters 16 (1972) 568; J.S. Hy& and D_D_Ti-mmas, Ann. E.Y.,Aczd. Sci. 222 (1974) 6ZO; M.D. Smigei, L.R. Dalton, J.S. Hyde and LA. Dalton, Proc. NatL Atid. Sci. US 71 (1974) 1925. [3] BH. Robinson, L.R. Dalton. L.A. D&on and A.L. Kwiram, to be pubtished. [4] B.H. Robinson,A.D. Keith,L.R.Dslton,L.A.CsIton and A+.. Kwiram, to be published. [5] F. Bfoch, Phys. Rev. 102 (1956) :04; R.K. Wangsness and F. Blo-_h, Phys. Rev. 89 (1953)
728;
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[6] A&. Redfieid, IBM J. Rer Develop. I (19.57) 19. [7] A; Abragam, The principtes OF nuctezr maggnetism (Clarendon Press, Oxford, 1961) chs. 8 and 10. [8] V.M. Pain, Zh. Eksperirn. iTeor. Fii 42 (1962) I.075 [English transl. Soviet Phys. JETP 1.5 (1962) 7431; V.N. Genkin, Fit. Tverd. Teb 4 (1962) 3381 [English transl Soviet.Phys.‘Solid State 4 (1963) 247.5’1. [9] R. Kubb;J. Phys Sot. Japan Suppl. 26 (1969) L; in: Stochastic processes in chemical physics. ed. K.E. Shuler (Wiley; New Yotk, 1969) p_ 101. [IO] J.H. Freed, A?. F?e\. Phys. Chem. 23 (1972) 265; in: Eiectron ,~in relaxation in liquids,‘eds. L.T. Muus and P.W. Attins (Plenum Press, New York. 1972) pp. 367-410, and references therein. [II ] A.L. Kwiram, I. Chem. Phys. 5.5 (1971) Z&34; : L.R. Dalton and &L..Kwimxn; J.Ckem. Phys. 57 (1972) 1132. .[12] .B.H. Rdbinson; L.R. Dalton. &.A. D&I and AL. Kw’iram, tobe published. 1131 J.S. Hyde, R.C. Sneed Jr.-.,i.R. Dalton and A.L. .Kw&am, to bcpubli&ed. .
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175 I “.
Volume 28, number 2
I.5 September
1974
THEORY OF MODULATION EFFECTS IN ELEC’RON ELECTRON DOiJBLE RESONANCE Bruce H. POBINkN; Departmkt
iean-Louis of Ciiemirtry,
MtiNGE*;
Luraine
Vanderbilt University.
A. tiA&TON,Larry
R. DALTON
T*
ffaslklle, Tennessee 37235, USA
and Alvin L. KIVIRAMi Department
of C?lemistry, University of Wcshington, Seattle, Washington 98195, US4
Received 7 June 1974 The nonlinear revonse of spinsystems to intense radiation fields is quantitatively treated by a modification of the stochastic Liouv-Ue equation for the spin density matrix. In particular, applied modulation terms are included in this equation. The resulting formal2.m provides a general method for calculating nonlinear spin response for dilute systems of idiuls in a high magnetic field. In this communication, frequency and field’swept absorption and dispersion electron-electron double resonance spectra are ukulate$ and compared with experimenta: spectra recorded under conditions of sinusoidal magnetic field modulation and phase-sensitive detection. Good reproduction of the detailed line-shapes of experimental spectra is observed in all casee. The dependence of ELDOR reduction factors upon modulation frequency is discussed. A theoretical analysis such as e,nployed in the present cJmmuni&tion is shown to be essential i.fELDOR reduction factors are to be,related to relaxation times and hence to molecular dynamics, and if the design of ELDOR experiments is to be optirmiied.
1: Introdl,ction
The response of spin systems to strong radiation fields has been shown to.be dependent’upon (1) spinlattice relaxation processes, (2) spectral diffusion processes which transfer energy throughout the reso; nzrce spectrum’(e.g., between resonant and nonresonant spin transitions), and (3) applied-magnetic field ., modulation of the resonance.condition 11; 21. Further-. more, it has been demonstrated [2] that nohear, methods can monitor sp:e&l diffusion processes for correlation times]significa#ly .long@rthan t&se for ‘. which i&h techniques are useful. In-particular, car-;:’ relation times over’s farige of four brders of nikgnitude centered
at the electron
spin-lattice-relaxation time using nonlinear ._ I_ :
(= 10y5 sj have been imxstigated .-
ESR as well as electron-eiedron daubk resonance (ELDOR) methods to study stable free radicals (spin labels) at low.temperatures [2J. For slowly tumbling riitroxide spin labels (molecular reorientation frequenties less than the frequency corresponding to tie magnetic,ariisotropy of 2rr X LO8Hz) ‘Lhedominant mechanism for spectrel diffusion is modulationof the anisotropic.electrori Zeemen and E4N hyperfime interktions caused by mole&r reorientation (tumblirig). For rapidly tumbling spin labels spectraI diffu-.
sion is determined by spin-flip processes -either nuclear spm-lattice, c+; processes or mtermolecular electron Heisenberg exchange, ; o,;;‘processes. For both the slowly and rapidly tuinblingregimes the applied magnetic field modula: .intramoiecdar
tjon p!aysa crucjal role in determining the character ‘-_ -.of the~nonliriear spin response. .’ : We l-&e &ce
.. ,
V&me 2,8,‘number 2 : .. ” _C,+;\IlcAL PHYSICS_-LETTERS “,_: -. ‘,_i 15 September 197.4 .’ .o..: ‘..,.. .‘.,. : :. : EUOR $nd E&response pfa spin system in the 1 ,- :,,. .’ ‘, ’ +j.= d; [s, e-?Jof+,s._ e’Wor] :sl& tti~ling reg,ir&:In t’lis commuqication we.+ .: p!;. a similr&formali&r to &cu!ate the r.espbnse for -iL+r,+$_ e&p’] + ds~z~[eiW sr+e-iwst] tdp[S+e .ra$dly?uqbling spin labels. For purposes o$ brevity j ‘. -an&ciarity we shall &strict our diskksion to simple ‘. _ : .mo’dels which illustrate most of the results of mole .’ + C iiiKizy [eiiuSr+e-iwsr] ,, ..(3) I( realistic (and more complex) model calcu!!ations : which we sh,all repoit :&-Idetai! eIsewhere ]4]f:The methods.outlined below provide ti powen% tool for _. where& =$yekti,,dp =fy~h,,d~=&,H,,and -the study of the correlation times associated with ’ $r,.,,H,. The frequencies of the’.microwave ‘IL =’ molecclar’diffusion, which gives rise to Heisenberi. .’ observer, microwave pump, and tieId modulation spin exchange or modulation of anisotropic interare designated by w,, , upI and os, respectively. The actions. fist tlwo terms on the right-hand,side of es. (3) describe the interaction ofthe electron spins with the ,’ microwave observer and pump fields; respectively. ‘. 2.,i;enek theory The last two terms describe the interaction of’the electron and nuclear spins with the magneticfield .To analyze the nunlinear response.of a spin system, modu:niion. The co,mmutator of the hamiltonian. f-lR to applied radiation and &odu!ation fields, we em-. which describes the coupling of the spin system to -pi.oy the general Boltznann equation for.the spin ,the la1 ticeis approximated by [7] density matrixas developed by, B!och [S J, Redfield -i[HR,o] =.-r~(u’u’)=-rRX ~ (4) [6j, Abragti [7], and others [8] and modSed tq treat the effects of markoffian rndtion by K&o 191 where go is the eqiilibrium gin density matrix. We and’Freed [lpJ:- . . express the’-off-diagonal matrix elements of this commutabzrr in terms of orientationahy invariant spin&, r) = -i[H(ti,tj,o(SI,t)] s-pin relaxation times and the diagonal elements are ex(1) pressed in terms of orientationally invariant spin:- :r,C& r)- uO(sr,t); ~ ,. lattice re!axation times. where I2 d&otes molecular orientation and )?a is a ,” The steady-state solution to (l), appropriate for time;independeni Markoff npemtor. Invoking the high frequency microwave fields, is obtained by using high field approximation (i.e,, neglecting terms : a series’expansion for X:. S,I,, etc. [l I]).the spin system can be described : by a syin hamiltonian of the forni H(R,
t) = ff,
t: H#k)
:+ c(r)
+, HR
:,
y,(ti,r)y (2)
= CZJQ,w7
a rw,)ei(~7*‘w~r
7
where the time-independent hamilt~oman H,, , which = p F”m
Volume 28, number 2
-CHEtiCAL
of ELDOR signals upon modulation;‘electron spinlattice relaxation, and upqn m&hanisms which transfe.r saturation between the pumped and’observed transitions, let us discuss the simplest system to .which ELGOR can be applied: a four-level system involving two electron spin transitions. We consider a particle described by a time-independent hamiltonian Ho = oPa S z J‘umpinop between two sites denoted by j= 1 or 2 and k= 2 or 1 . Fbr the two-site jump model we have, considering only secular terms, H,Q)=FG)S, where F(1) = -F(2) = F. F is one-half tlie total magnetic anisotropy or frequency difference of the magnetic interactions in the two sites. The operator in eq. (1) whichdescribes the markofsan mction is
r=
-w,
.(
WJ
y-w;
WJ ) ’
15 Se&tber
PHYSiCS LEFERS
+Jiw,Y,(i,Frw,) = dp {Z,,,(k,
1971
(74
wp +-rwJ - ZJ”,
--wpfrw,)
5,
where A, = w. - wpa and q = fi/kT. We have emFlayed the additional definition that the diference between the diagonal elements of the reduced density matrix is given by,
(6)
where oJ denotes the mean jump rate. Given the above expressions for f-lo, HI, and I’ and denoting the two electron spin states as usual by a and B we express (1) in terms of the matrix elements Zga using (5). This ;esults in the iollowing set of general coupled equatioris for the pumped and observed transitions:
Information regarding the magnitude of the respoase. of the spin system resides in the (complex) matrix elements ZpO and we must solve the simultaneous equations in (7) for these matrix elements. However, the Zpcl are not themselves the experimental observables. Indeed, the design of the microwave detectioncircuit together with the use of a phase-&nsitive detector results in the selection of a particular Fourier amponent of the spin-system rer,ponse given implicoutitly .by Z,, e iwf, In general terms, the utitered put of the phase-sensitive detector is given by’
+~~~(~,~rw,)e-~(~~““~)‘~
sin worcos(rws~+
‘.
(9
where the first factor describes the coherent response of-the spin system to the applied time-dependent‘ field and the second factor represents the reference ‘. sgml employed to select a given Fourier component of the resFon=. Thus the microwave sykem is adjusted TOselect either the dispersion signal (coefficierit of the cos uof term).or the absorption signal
=$This four-level model al_~ descibes intermolecular Heisenberg’spin.exchange between two’electr6n sini : (wex processes) as yell as the,gLDOR‘ie@nse for a. coilpled electron-nuclear S=I= l/2 -fin sys,tem domk ‘kted by intramolecular nticlw spin relaxation (wn
procqes).
@I] ) dr;
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‘CHBMKALPHYSICS
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(coefficient ofthe’;in w,t tekj, Then rhe phasesensitive detector is adjusted to select the in-phase
such as, for example,
Z(1) *Z(2)
and r(?)
= Y(1) + Y(2).
4. Results and discussion .’
Proceeding from (7) we have numericalJy computed the various l%DOR signals [(combinations of Zpao(+, o_-Ot rwS) and Z&(+, w. - rws)] at the first and second harmonics of w, for coupling to barmom&as high as f = 40 under the condition that the pur+observer’frequency separation is equal to the exact frequency separation between th.e two resonance lines, i.e., I-up - woI = 2F in the case of field-swept spectra and under the condition that w,, = wBu + F in the case’of’frequency-swept ELDOR spectra. The -calculated line shapes.for the latter case are presented in’fig. 1. The agreement with’experimentally observed line shapes is excellent. Likewise, excel!ent agreement is obtained for field-swept ELDOR spectra+: In addition to-the Xne shape information we’also obtain explicitly the’dependence of the ELDOR signal amplitude,.ususlly~expressed in terms of the re-. duction factor, on instrumental and molecular variables. The: ELDOR reduction factor is defined as R = I 7 Izp~o(~;
= ‘WZ9(ro(dp
= 0))
Fig. 1. Computed frequency-swept ELDOR spectra at the first axd second harmonics of the modulation and in phase with tte modulation a.re shown. (A) First harmonic absorption; (13) fast harmonic dispersion; (C) second harmonic absorp-ion; (D) second liarmonic dispersion. Parameters employed in the calculation include (w -‘wo)/Wo = +F/do =’ 2000; 42d, =0.63;d&io= 2;,WJ 72d, - l.l4;4d&T,Tz = 0.062; Tl = 2Tz ;,dp/do = 5; w. = WB - F. Spectra shown represent plot of signzrl (abitrary units) versus frequency separation of pump and observer, with frequency increasing to-the right. ELIJOR Lines are centered at wp = w& + F.
where Zpoo(dp = O), the ESR signal pump off, represents a combination such as ~z;~o(:+. G.‘o+ rw,> + z;,,
the in+ase dispersion signal. Many variables such as .h,, h,: T,, T2, uJ, w,, etc., influence the reduction factor: we limit the discussion to the more commonly encountered experimental situations. Representative behavior of calcuiated ELDOR reduction factors is presented mfig. 2. For low modulation amplitudes the effect of modulation frequency on the in-phase dispersion signal is insigni5cant: However, the-commonly measured absorption signals gre ser&tive to modulation frequency: the ELDOR reduction factor decreases
,’
..
(+, 00 - rw,> 1 2
chA&‘iriW~
increas?
k&w,
T1 (cf. fig--Z),
t& fmcdond char& ii i.nipp+nt even for mJ T1 4 ) It can be shown that thk mlcuiated lineshapes &equally ‘. 1. Nofe too that the reduction’faciors for frequencyvalid Pol.dilute free radicals in the soid state. Therefore, : swept absorptionBLDOR can be significantly less these rcsuits (;1n be used for th6 analysis of ELDOR spe$Fa 1. of randdmly oriented radic&s in the solid phase:. &~n’~th: reduction factors for field&ept absorption ‘. “.’ ‘, .; ‘. IT’, -, .1 : : .’ /, ‘> .. ‘-, ,: .,,‘. ..::..., :’ : j’. .. : : _’ ,, ;,:; _, _,., ,_... ” : I . . . . -. ., ‘.: ...I ... : -:. -.. : _:- : .,,‘.’ ..‘.,; ,. .I .,.. ., ,. .. ,‘. 1:. .:.’ ‘. .:.: ..‘. _: ,‘, : ..,. .” : .“_ -:’
0.05
15 September
CHEMICAL PHYSICS LETTERS
Volume 28, number 2
05
UJTl
Fig. 2. The dependence of the ELDGK redukn factor for various in-phase signals upon WJ T1. The open snd solid circles represent the reduction factor for the field-swept (up- w. = 2E) absorption ELDOR signal at the first 11xmanic of the modulation for a value of wsTl .= 0 and for ws5”l = 3, respectively. The dashed line (solid t&gles) IO presents the field-swept dispersion ELDOR signals at the fLrst harmonic of the modulation for these modu!ation frquecties. The open squares represent the frequency-wept absorption ELDOR signal at the first harmonic of the modulation with the observer positioned at w. = apn - F and with w,TI = 0. Othei parmeters utilized in the c&ulation include; F/2d, = 1000,4cI~ TITz = 0.07, Tl = 10Tz; and dp/do = 8.
ELDOR, as is experimentally observed. Also in agreement with theory, we have observed that dispersion ELDOR reduction factors measured for field-swept and frequency-swept &odes are identical. The decrease in reduction factors at high WJ TI arises from the spectral diffusion process shortening the effective T1 and T2, making it more difficult to saturate the observed transition; This effect can be visualized by noting that.in the limit of no modulation we can write the saturation factor as
where n = 2, n! = 1 for the two-jump modei and 12= 3, m = 2 for a thee-jump model (or equivalently for Heisenberg spin exchange coupling the three electron ridiwl). spin transitidns of a nitrotide A point of considerable practical significance evident from the coniparison of reduction factor’s fbr various operational modes is thar’ dispersion ELDOI? &fords significant advantages relative to absorption
1974
ELDOR r&t only.in the ineasurement of energy -.r transfer rates (wJ , w,, c+,) but also in yieIding better signal to noise in the measurement of spectroscopic parameters (hyperftie intera6tiot-G). For this litter purpose, ELDOR is usually performed in *-hefrequency-swept mode by positidnitig the observer at exact resonance for some Gansition. Our theoretical and experimentaYresuIts confirm that better sensitivity is obtained in dispersion mode for rapidly tumbling radicals (and for radicals in single crys!als). The dependence ofR upon $/k;is shown in fig. j for two values of ho. For values cf It, in the linear spin response region (fig. 3A) the general slzope of the reduction factor curves is independent of fl, with the curYes merely being displaced aIong the $,jtl, axis for differen t values of ho. AIso if the saturation factors [see eq. (1 l)] for two- and three-jump models are adjusted to be equal, the calculated curves of R versus ~zJ~, for these two modeIs are virtually superimpo-able. The dependence oFR uporr h,lh, is quite insensitive to modulation amplitude as long as the modulation amplitude is substantially less than the spin packet linewidth; however, as the modr-dation amplitude approaches the linewidth, significant effects are observed. By comparing a series offigures such as those shown in fip. 3 it is readily seen that the greatest effect of modulation frequency on the ELDOR reduction factor occurs at precisely those conditions under which most ELDOR measurements will be made, i.e., near the linear response region on the observer sat&ation curve and for pump-observer ratios between 2 and 20. These results clearly demonstrate that modulation effects play a crucial role in determining R acd cannot be ingored (as has been past practice) in relat,ing ELDOR reduction factors to molecular energy transfer rates.(in .&is case to ws T1 ). It should be emphasize2 that by appropriate replacement of oJ by w, @Wex our calcuiations aIso reproduce the effect of intramolecu!ar nuclear spinlattice relaxation or intermolecular Heisenberg spin exchange for a four-level system. Extension of the calcl$ations to *sterns with mbre than four levels is st+i&tfo&=rd. more
Furthermore,
genera1 spectral
Hfusion
the intrcduction mod&k
readily
of ac-
cornmodated by the fo,n@lism discussed herein. To illustrate this we have &ied out extensive calculations for a:typica:six-level system (S =. l/2,:1 = 1)
..volume
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28, number 2 ..’
CHEMICAL PPFiSkS
LEl-IEti
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.J5 September
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,. Fig. 1. The
Qlcuhted variation of the ELDOR reduction faGfor_.. .for field-sa.ept absorption
and dispersion ELDOR at the fust harmonic OKthe modulation and. in ph.ax with the modulation versus the rati) $/ho (or, equivalently, +/do).. The circles and ‘quarkrepresent ihe ibso&ionR’s calcr~lated for ws’i’r = 0, and for wsTr = 3, respectively. The dashed line represents the disper_iion reduction faetors;,which axe apprdximetcly independent of &,Tr _(A) is cimputed for a value of ho in the hncar spin response region (no satluatinn) whi!e (B) is computed for’xwlue of ho in the nonlinear region (hard saturation) of the observer mturatidn cuNe. Other parameters used in the c&&ion are T1 = Ta, WJT, = 2.5, F/ado _ = 1000.
:
appropriate for a nitroxide radical. These calculations take int s account both w, and sex spin-flip proces‘ses. Moreover, the two-site jump model has been replaced
by art isotropic
brownian
diffusion
model
in
order tc reflect more realistically the character of the motion which modulates the anisotropic magnetic inteMctions [12]. In fig. 4 we present the theoretical simulation (crosses) of a typical frequency-ssvept ELDCR spectrum for a nitroxide radical in a low .viscolity solvent. The calculated value for the product of the molecular relaxation paramet.ers is We,T1 s 3, in reasonable agreement with the value expected theoretically for this system at T = -22~5°C. In order to illustrate an alternate means of.deterI I I mining the molecular relaxation parameters we re20 40 60 ‘80 produce in fig. 5 representative ELDCR reduction -. fYp-90) in MHz . ., factors measured at modulation frequencies of 270 Fig. 4. The experilyental in-phase disp,zrsion ESR (A).at the Hz and IO0 H-Iz for absorption and dispersion detecfirst harmonic of the modulation’and the in-phase dispersion tion. The theoretically-predicted diminution of ab.’ELDOR (B) at the first harmonic recorded in a frequency sorption reduction factors with increasing modulation srieep mode with the microwave observer at the exact re&-: frequency is obs’erved. When a least squares fit of the nanceofthcm~=-1 r4N hyperfine Line. The sample is a experimental data to theoretical curves is carried out, 3 X 10-a M solution &2,2,6,6-tetramethyl+piperidinol-l.oxyI (TANOL) in set-tutylbenzene .at -22.S°C..The follow.all sets cf data extrapolate to the same R(dp +.-) ing conditions apply: So = O.OOz?rad Hz. ws = 6.28 X IO! value, as.is theqretically required. This value is 0.8, m-d Hz, dp (=0 for ESR. = 5do for ELDOR). The fist index’ ii:hichyields ,for the product of molecular relaxatipn, in the parentheses associated with R indicates the nuclear with the :$Qn quantum.number of the-14N hyperfme hne pumped ‘1 ,. paran’efsr? we.x.Tl = 4; ag~~:cor&t&t ,while ihgsecon3 indicates the.line observed, The crosgs ‘., ,valuis e::pected:for- this system ,at T = Sa,Ci The two r@r&nt theoretk+hy c&r&ted vahks and the’best fi,t car- ,:’ methods illustrated in figs.4 and 5 provide in impor- ‘. -iespo+dsto &.&Fl G.3:. .:’ : ., :‘,‘, .. ~ : -. .. :. ‘j$4”‘_,:’ ..I +._ I
,
!
Volume 28,‘number
C~MI&L
2.
PI?YSICS LETTERS
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IO
hi/h,
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1974
A.&no@edger~ent is’made to the Donors of The Petroleum Fesearch Fund, administered by the, Aiierican Cl?p_mi&al Society, tj the Research Cosporation, and tq the National Science- FoL:ndation @P-28183X arid GP42998XJ for p.zrM support of t?b research. We acknowiedge suppoti for the construction of nonlinear spti response instrumentation from the Cherr&al Inatrumentztion Section, National Science Foundation (GP-3.6774).
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1.5 September
“‘.
Fig. 5. The variation of experimental field-swept in-phase absorption and dispersion ELDOR redwction fact&s; R(O,+l), detected at the first harmonic of the rno~u~tion versus +/ho (or, quivzlently, +/do). The absorption reduction factors are shown for twd moduhtion frequencies; name@ ws = 6.3 x 10’ rad Hz (circ!es) and ws = 1.7 x f03. rad Hz (squares). The dispersion ELDOR reduction factors, denoted by triangles, were recorded at a modulation frquency of 100 kHz.‘The s-ample is a 3 X 10e3 M solution of TANOL in secbutylbenzene at 8°C. The.solid and dashed lines z& best fit theoretical values calculated for So 4 0.005 and wexTl = 4. The foLIowing magnetic parameters were also employed iE the calc&tion: gxx = 2.0094, gjlv = 2.0961, gzz = 2.0321,g = 2.00587.d,(‘4N) = 18.248 M.Hz.A~~(~“N) = 18.810 hfij~,A~~(‘~N) = 92.646 MH~,F(‘~N)_=.43.235, hil-l~!~Si(~H) = i.35 MHz.
taut test of the theory and in all cases thk results have been consisterit. It is important to note that the ELDOR reduction factor plots contain no adjustable parameters. In order to niake meaningful comparisons of experimentaland theoretical reduction factors one must have precise values for the ampIitudes and phases of all the radiation and modulation fields over the s&nple volume. We have expended considerable engineering effort in cqntrolling and pieeisely measuring these factors. Full details of this work, together with a description of our ELDOR spectrometer, which pas.- sesses ~pab~ities for absorpt,ion and dispersion ELDOR measurements at .the first and secqnd ha& mimics of the moduIation and tit twer-$y’discrete’ modulation frequencies ranging from 35 Hz to ! MHz, will. be presented elsewhere 1131.
[1] L.R. &tori, A.L.Kwiram and J.A. Cowen: Chem. Phys. Letters 17 (1974) 495. [2] J.SI Hyde and L. Dalton, Chem. Phys. Letters 16 (1972) 568; J.S. Hy& and D_D_Ti-mmas, Ann. E.Y.,Aczd. Sci. 222 (1974) 6ZO; M.D. Smigei, L.R. Dalton, J.S. Hyde and LA. Dalton, Proc. NatL Atid. Sci. US 71 (1974) 1925. [3] BH. Robinson, L.R. Dalton. L.A. D&on and A.L. Kwiram, to be pubtished. [4] B.H. Robinson,A.D. Keith,L.R.Dslton,L.A.CsIton and A+.. Kwiram, to be published. [5] F. Bfoch, Phys. Rev. 102 (1956) :04; R.K. Wangsness and F. Blo-_h, Phys. Rev. 89 (1953)
728;
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[6] A&. Redfieid, IBM J. Rer Develop. I (19.57) 19. [7] A; Abragam, The principtes OF nuctezr maggnetism (Clarendon Press, Oxford, 1961) chs. 8 and 10. [8] V.M. Pain, Zh. Eksperirn. iTeor. Fii 42 (1962) I.075 [English transl. Soviet Phys. JETP 1.5 (1962) 7431; V.N. Genkin, Fit. Tverd. Teb 4 (1962) 3381 [English transl Soviet.Phys.‘Solid State 4 (1963) 247.5’1. [9] R. Kubb;J. Phys Sot. Japan Suppl. 26 (1969) L; in: Stochastic processes in chemical physics. ed. K.E. Shuler (Wiley; New Yotk, 1969) p_ 101. [IO] J.H. Freed, A?. F?e\. Phys. Chem. 23 (1972) 265; in: Eiectron ,~in relaxation in liquids,‘eds. L.T. Muus and P.W. Attins (Plenum Press, New York. 1972) pp. 367-410, and references therein. [II ] A.L. Kwiram, I. Chem. Phys. 5.5 (1971) Z&34; : L.R. Dalton and &L..Kwimxn; J.Ckem. Phys. 57 (1972) 1132. .[12] .B.H. Rdbinson; L.R. Dalton. &.A. D&I and AL. Kw’iram, tobe published. 1131 J.S. Hyde, R.C. Sneed Jr.-.,i.R. Dalton and A.L. .Kw&am, to bcpubli&ed. .
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175 I “.