longitudinally modulated endor spectroscopy. ii. variable frequency measurements

JOURNAL OF MAGNETIC RESONANCE 84,309-322 (1989)

Longitudinally Modulated ENDOR Spectroscopy. II. Variable Frequency Measurements

CALOGERO PINZINO

Istituto di Chimica Quantistica ed Energetica Molecolare, C.N.R., Via Risorgimento 35 ,561OO Piss, Italy

Received September 22, 1988

A new technique allows the fongitudinaliy modulated electron nuclear double resonance (LOMENDOR) signal to be observed at fixed static magnet ic field and microwave frequency, by sweeping the f requency difference between the two irradiating radiofrequency waves. The experimental setup is described. The linewidth dependence on the “cross” relaxation time T,, is discussed. The method is suitable for nuclear longitudinal relaxation time measurements. The dependence of the l ineshape of first and second harmonic signals on the f requency difference and on the electron and nuclear saturation factors is also analyzed and discussed. 0 1989 Academic press, hc.

INTRODUCTION

In a previous paper (I) a new technique, named longitudinally modu lated electron nuclear double Yesonance (LOMENDOR), was described. An electron and nuclear spin system placed in a static magnetic field B. is transversally irradiated by one m i- crowave and by two radiofrequency electromagnetic waves. The angular frequencies of the two RF waves, o, and o,, are close to the same nuclear magnetic resonance frequency and the angular frequency of the m icrowave, w,, is close to the electron magnetic resonance frequency. Then the transverse components of the electron spin system magnetization, J4, are modu lated by the oscillating longitudinal component of the nuclear spin system magnetization A4,, . The longitudinal component M ,,, has oscillating components at every harmonic of the frequency difference 1 w, - w, I.

The LOMENDOR signal amp litude, obtained from the detection of the oscillating components of M ,., depends on the relaxation times Tze and T,,; on the relaxation rates W ,, W ,, , W ,, , and IV,,; on the amp litudes B,,, Bf’, Br’, and By’ of the static and oscillating magnetic fields; and on the values of w,, wr, and w,. The principal variables determining the LOMENDOR l ineshape are the three quantities A, = w, - w. - A,/2, AL = w’ - w, - AJ2, and 6 = w, - w,, where w. and w, are the electron and nuclear Larmor frequencies, w’ is half-sum of the RF frequencies and A, is an effective hyperfine coupling for a particular orientation of the static magnetic field with respect to the axes of the hyperhne interaction. Since A,, A;, and 6 can be varied independently, three different methods may be applied to LOMENDOR spectroscopy. In the first (LOMENDOR) and second (LOMESR) AL and A,, respectively, are varied, keeping constant all the other quantities. In the third, hitherto unreported method which we call variablefiequency longitudinally modu lated electron nuclear

309 0022-2364189 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.

310 CALOGERO PINZINO

double resonance (VF-LOMENDOR), & and Ah are fixed while 6 is varied. In every case the harmonic expansions of the electron transverse components and the nuclear longitudinal components of the spin system magnetizations may be written

h4,, = $J &exp[iw,t]exp[ik(w, - w,)t] + complex conjugates k=O

Dal

M”Z = c PkexP[ik( W, - u&] + complex conjugates. [lb1 k=O

We have recently examined the lineshape (1) related to the Lk coefficients as a function of the A; and A, parameters. In particular we showed that at low saturation the LOMENDOR and LOMESR signals corresponding to the L, harmonic have a linewidth depending only on the nuclear and electron transverse relaxation times, respectively. The L1 amplitude is proportional to a “cross” electron and nuclear relaxation time T,,. Furthermore we showed that the measurement of the ratio between the second and the first harmonic signals allowed the determination of the nuclear longitudinal relaxation time.

In this paper we report the main features of VF-LOMENDOR signals. The signal linewidths, at low nuclear saturation factor, depend on both relaxation times Tzn and T1,. Nevertheless, the dependence on the relaxation time T,, is sometimes the more significant one. On increase of the electron and nuclear saturation factors, the lineshape exhibits a characteristic maximum whose position as a function of 6 depends on the microwave and RF irradiation intensity. Furthermore we show that the signal saturation is described by two saturation factors S, and S, which depend strongly on the 6 value. We then describe the experimental apparatus, which is based on that of Ref. (1) but which required extensive modifications in order to improve the frequency range and allow continuous sweep of the frequency difference 6. The experimental results are shown to be in good agreement with the numerical and analytical predictions of the LOMENDOR theory.

THEORY

The theory of lineshape and signal dependence on relaxation times in LO- MENDOR spectroscopy has been presented recently (I). For the full definitions and for all the detailed calculations to obtain the set of the used equations governing the LOMENDOR signal lineshape the reader is referred to Ref. (I).

We treat an S = 1, I = 4 spin system when irradiated by three electromagnetic waves. A density matrix p which satisfies the Liouville equation generally describes the spin system. This equation can be written as (2)

(i/I’)D = [x, D] - P[%Ic, prl/Tr{exp - P(8s + RI)} PI

if the operator p is written as p = p. + D, where p. is the density operator at thermal equilibrium and D is the part of p which is affected by the relaxation processes. In Eq. [2], I is the relaxation superoperator; 2 = zs + zR + xi, where zs is the

LONGITUDINALLY MODULATED ENDOR SPECTROSCOPY 311

Hamiltonian of the spin system, ZR the Hamiltonian of the radiation fields, and 2, the Hamiltonian of the spin-radiation interactions; p is the Boltzmann factor.

In order to solve Eq. [2], the oscillating magnetic fields are described with the second quantization formalism and the observables of the spin system plus radiation total system are represented on the basis of the eigenstates

I me, m ,, 4, n,, n,), where m ,, m , are the electron and nuclear spin z-component quantum numbers and n,, n,, n, are the photon occupation numbers for the three electromagnetic fields.

Since the interaction energy between the RF radiation and the nuclear spin system is small in comparison with 1 Zs + ZR 1 and the three waves are nearly resonant, the eigenstate manifold will assume a quartet structure and we can order it in the succession

... I+i,+f,n,+ l,n,+ l,n,-I), I +1, -1, n, + 1, n,, n, - l),

1-$,+4,n,,n,+ l,n,- I>, I-1, -1, n,, n,, 4 - I),

I+i,+t,n,+ l,n,,n,), I +$, -1, n, + 1, n, - 1, n,), I-t, +f, k 4, ns), I-t, -4, n,, 4 - 1, ns). I+$, +$, n, + 1, n2, - 1, n, + 1), I++,-i,n,+ l,n,-2,n,+ I),

I-t,+t,&,nr- I,%+ I>, I-t,--~,n,,n,-2,n,+ l),.. . . [3]

These states are labeled as . . . -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, . . . , the label 0 corresponding to the state I +$, +$, II, + 1, nr, n,).

A statistical average (I, 3) yields for the kth harmonic coefficients Lk and Pk of Eqs. [II

Lk = 1/b’eNe(&,4k+2 - hk+4) W I pk = 1/b’&(&,,k - %k+l)> [4bl

where the matrix elements of the operator D are represented on the basis labeled as in [ 31; -ye, Y,, are the electron and nuclear gyromagnetic factors and N,, N, the numbers of the electron and nuclear spins for unit volume.

Reference (I) shows that when all off-diagonal elements of the D matrix, except those relating to the irradiated transitions, are set equal to zero and the recurrence rule Di,j = D,+&,+,+k is taken into account, the master equation [2] for the state solution of the operator p leads to the following set of linear equations involving the elements LI, and Pk,

Lk = u,P, + hk PaI [I - sr’(ok-, + 0,)/2]Pk = s~‘o,-,P,-~/2 + $%kPk+& + ck, I5bl

where

uk = -iX,T2,(l - ik6T2e)TE’/(GkT$,\Rk)

R,, = 1 + T:,g + Sy’( 1 - ikST,,) - ik&T,,(2 - ik6T2,)

[5cl


Sl.“’ = 4e Tze T$, se = go’

Sr) = Sa’Gk = 4X2T2, T($Gk, S” = s’,“’

Gk = 1 - A,’ T,,( 1 - ik6 T,,)( T$‘/(Rk T$)

hk = 0 if Ikl #O

ho = -i2bX, T2,w,/R’

R’ = 1 + T&A,2 + SJl - T:,/(4T,,fT,.,)]

0, = [i - TznAh + (k + &)6T2n]-’ + [i + T2,,Ah + (k + i)6T2,]-’

and

[W

ck=O if Ikl +O

co = -ib2Xz T2,T,,w,/( 1 + Tie@ + Se).

In the above expressions A,, h are given by

A, = y,B1”‘/2; X = 7nBd2,

where By’ is the amplitude of the microwave magnetic field and Blf = B, (1 + A,/ (2w,)] with B, = B’,” = By’ is the effective amplitude of the magnetic fields (4); Tz,, , T2e are the nuclear and electron transverse relaxation times; and T’k:, 7’(/$, and T’z are given by

T’i{ = T~~(2W, + W,, + W, - ik6)/(2W, + 2W, - ikS) [6al T’:,‘f = T’:,‘(2W, + We + W, - ikS)/(2W, + ZW, - ik6) [6bl

and T@’ = IP 1/(2W e + 2W ” - ikF). [W

Here, for simplicity, we have assumed the cross relaxation probabilities W,, = WX2 = IV,. Equations [6] give, fork = 0, the “longitudinal” electron relaxation time Tlef, the “longitudinal” nuclear relaxation time TLnf, and a “cross” electron and nuclear relaxation time TIP (5).

The set [5] can be solved analytically for simple problems but, if several harmonic signals must be studied at high irradiation intensity, the equations require numerical solution. To this purpose a computer program has been developed which solves the set [ 51 within the proper approximations including all processes involving up to some chosen number of photons.

To obtain analytic solutions of set [5], we must first solve Eq. [5b] perturbatively in order to obtain P, . If we assume a low level of RF irradiation and if we neglect processes involving more than three photons, it is straightforward to show that

lPll = IC&‘([l +(T,,6)=/4]“*/F’, [71


where

F’= {[l - (7’2n6)2/4 + (7’2nA:,)2]2 + (T2n~)2}1’2.

From [5a], taking into account Eqs. [%I, [5d], [6c], [7] and assuming Tze8 4 1, we obtain the correct second-order solution for the first harmonic signal 1 L, 1

IL, I = QSn[l + (T2nW)21/{[~ + (T$)*1”*~‘) @I

with

Q = 2~~(LT,,)2uJ(& IRI I Td

The coefficient 1 L, 1 gives the lineshape of the first harmonic LOMENDOR or LOMESR signals as a function of the AL or A, parameters. Equation [8] shows that the amplitude and linewidth of the signal depend on 6. In particular, the LOMESR signal linewidth can be approximated by

(A&* = a{-v + [v2 + 4(v - 1 + uS,Z)]"*}~'*/T~,, [91

where

0 = 2(U + l)S,, u = [l + (T,,6)2]-“2{[1 + (6/llQ2]/[1 + (6/W2)2]}“2

withW,=2W,+WX+W,,andW2=2W,+2WX. IfTr,6< 1,wehaveum I and the linewidth is given by

@eh,2 = 2u “* - l)(l + Se)]"2/T2c. WI

The same result was obtained in Ref. (I) within the same approximation. On the other hand, if T,$ $ 1, we have u = 0 and the linewidth is

(A,),,, = (1Jz/T2,>{-(2 + Se)+ [(2 + Se)* + 4(1 + ,,]“2}“2. Pbl

The LOMENDOR signal reaches its half-amplitude, at low values of the electron saturation factor, when

/A,11,2=(1/T2,){-l +(T2,S/2)2+2[1 +(T2,6/2)4+(T2nS/2)2]"2}"2 [lOa]

or

ifwe let T2,,6 -@ 1.

I A, II/~ = l/T,, [lob1

Now, if we assume A, = 0 and A; = 0, the coefficient I L, I becomes

IL, I = G?W{[l + (T,,@21[1 + (T2n~/2)21)“2 [Ill

which gives the lineshape of the first harmonic of the variable frequency LO- MENDOR signal. The coefficient IL, I has a maximum when 6 = 0 and, if S, < 1, reaches its half-amplitude value when

16 I 1/2 =~(T~T&ITI~ [121

314

with

CALOGERO PINZINO

f(T2JTlp) = { [-( 1 + z) + (( 1 + z)’ + ~sz)“~]/(~sz))“~,

where z = [T2,/(2Tl,)12 and s = (3 + 8 5’,)“2. For z < 1 we have

161 l/z = 11 - U’~,/~I,)~I”~(~ + 8 &)“21T~p

which, for T2,, / T, ,, < 1, can be approximated by

16 1 ,,2 = ti( 1 + S/3Se)“2/T,,. [I31

Equation [ 121 shows that, at low electron saturation level, the linewidth of 1 L, 1 depends only on TIP. In the general case, sincef(T2,/T’,) is a slowly varying function in the range of interest (0 < T,,/T’, < 1) (6), the dependence on T,, is by far the main feature of the first harmonic signal linewidth.

From set [5], by neglecting processes involving more than five photons, the coefficient of the component A4,, oscillating at the frequency 26 can be obtained,

IL2 1 = Qq&{[l + 5T:,6* +4T:,64][1 + 5T:,cS2/2 + 9T;,64/16]}-“2, [14]

where q, = {[1 + (6/W3)2]/[1 + (6/W4)2]}“2 with IV, = 2W,, + IV, + W, and IV4 = 2 IV,, + 2 W,. Equation [ 141 is obtained for the case of variable frequency LO- MENDOR signal (A, = 0, AL = 0) at low electron and nuclear saturation factors.

The ratio, obtained from Eqs. [ 1 I] and [ 141, between the second and the first harmonic signals

IL2 l/IL, I = Sn{[l + (2T,,6)2][l + (3T,,W)2]}P”2q, [I51

shows strong dependence on the frequency difference 6. Finally, if we neglect processes involving more than 2k photons, we can obtain in

the same approximation the lineshape of the kth harmonics

1-h I = Sn IL-, I {[I + W’,,S)21[1 + (k- t)2(T2n~)21}~“2qk-,, 1161 where qk = { [ 1 + (k~5/W,)~]/[ 1 + (kS/W4)2]}“2. Equation [ 161 shows that, increasing the k value, the lineshape of the variable frequency LOMENDOR signal strongly narrows.

EXPERIMENTAL

Instrumentation. The LOMENDOR spectrometer, used in (I) to observe the oscillating components of electron transversal magnetization, has the following draw- backs which do not allow experiments at continuously variable frequency. First, due to residual FM deviation of the swept oscillator used to generate the second RF wave, the minimum frequency difference cannot be reduced below 1 kHz. Moreover, since the two RF generators are phase locked, an asymmetric shift with respect to the nuclear magnetic resonance frequency results in consequence of the change of the frequency difference.

LONGITUDINALLY MODULATED ENDOR SPECTROSCOPY

FIG 1. Block diagram of the LOMENDOR spectrometer modified for variable frequency experiment.

To overcome the above difficulties, in the present experiments a single RF generator has been employed (Fig. 1). The two irradiating waves, required to obtain the LOMENDOR effect, are produced by multiplying the resonance frequency w’ by a signal at frequency Q which can be only swept with continuity in the range 2.5 Hz to 100 kHz because of lock-in amplifier (EG & G 5208) requirements. The swept signal was generated by an HP 3325A synthesizer generator. The frequency spectrum at the multiplier circuit output consists of two main sidebands w,, w, which are symmetrical with respect to W’ and whose frequency difference 6 equals 2fi, plus some signal which is down by at least 40 dB. Since signals at frequency 20 must be detected, it was necessary to connect a frequency doubler between the output of the sweeper and the external reference channel of the lock-in amplifier. The frequency doubler consists of a phase comparator, a linear voltage-controlled oscillator, and a frequency divider. Due to the lim ited frequency range, the HBF 4018A (presettable divide-by-IV counter) and HBF 4046 (phase-locked loop) cos/mos integrated circuits could be used.


The remainder of the apparatus (7), consisting of an E- 112 Varian Century Series ESR spectrometer, an HP 8660B frequency synthesizer, and an EN1 3 1 OOL RF power amplifier, was exactly as that described for the LOMENDOR experiment (1).

Methods. To obtain variable frequency LOMENDOR measurements the procedure is the following. First the ESR spectrum is scanned by sweeping &, . Then &, is positioned at the crossover of the hyperfine line to be used. The apparatus is then switched to the LOMENDOR mode. The next step is the establishment of the frequency difference 6 between the two RF waves by the variable frequency drive of the sweeper. Then the LOMENDOR signal can be detected and recorded by sweeping 0’. A sequence of LOMENDOR spectra at different 6 values can thus be recorded. The VF-LOMENDOR spectrum results by plotting the maximum amplitude of LOMENDOR signals.

Due to a particularly attractive feature of the used lock-in amplifier, which main- tains reference lock as the reference frequency is changed, another simpler technique could be used. If we set the RF frequency w’ at the value that produces the maximum LOMENDOR signal, the VF-LOMENDOR spectrum can be obtained by simply sweeping 6 in the useful frequency range.

RESULTS AND DISCUSSION

Investigations have been performed at room temperature on the a-proton LO- MENDOR V+ line at 28.165 MHz of a single crystal of malonic acidurea (MAU), which forms a stable radical upon r-irradiation (8). This line was obtained, as in Ref. (I), by positioning the single crystal so that the X principal axis of the hyperfine interaction was along the direction of the static magnetic field and the Y one along the direction of the RF fields. So the very values of the relaxation transition probabilities W, = 3.2 X lo3 s-l, W, = 1.86 X lo3 s-‘, and W, = 80 s-’ must be used in all the numerical calculations.

The electron and nuclear relaxation times Tze = 1.90 X lo-’ s and Tzn = 2.85 X lO-5 s have been determined, from Eqs. [9a], [ lOa], by measurements of the half- width at half-amplitude of both LOMESR and LOMENDOR spectra, far from saturation.

In Fig. 2, we show the dependence of the linewidth for first harmonic LO- MENDOR and LOMESR spectra on the frequency difference 6. In fact Fig. 2 shows the quantities W, and W,, defined as the ratio of the above linewidths to the unsatu- rated first harmonic linewidths, against 6. The solid lines represent the results of the analytical calculation (Eq. [lo] and Eq. [9], respectively), which agree well with the experimental results. It should be noted that the linewidth of the LOMENDOR spectrum shows gradual broadening as 6 increases, and on the contrary the LOMESR linewidth shows narrowing.

To investigate the LOMENDOR signals as a function of the 6 value, the L, amplitude is reported in Fig. 3 at different nuclear irradiation levels and at a low S, value. The numerically obtained lineshapes, computed from set [5] by neglecting only terms corresponding to processes involving more than 12 RF photons, reproduce well the experimental results under all conditions, whereas, on account of the used approxi-


I I / I I I / 1 I

5 +104 Hz)

10

FIG. 2. LOMENDOR (IV,) and LOMESR (IV,) linewidths versus b for a protons in a MAU single crystal. W, and W, are in units of 2/T,, and 2/Tge, respectively. The saturation factors were S, = 3, S, = 0.03.

mations, the analytical predictions [ 1 l] (dotted lines) agree with experimental observations (triangles) only at low S,, values. The curves show a sharp maximum centered at 6 = 0 with linewidths very close to the values obtained from the analytical expres-

Ll’

2

5 10 56 ~(104Hz) 10

-6

Ll

- 4

-2

FIG. 3. First harmonic frequency-swept LOMENDOR signals as a function of 6. Electron saturation factor S, = 0.08; nuclear saturation factor: (a) S. = 0.02; (b) S, = 0.05; (c) S, = 0.1; (d) S, = 0.26; (e) S, = 2.11; (f) S, = 5.32. Solid lines: numerical solution of the set [5]; dotted lines: approximate Eq. [12]. Triangles: experimental results for the (Y protons in a MAU single crystal. All quantities are in the same arbitrary units.


27

% I!J

&IO4 Hz)

FIG. 4. First (upper lines) and second (lower lines) harmonic frequency-swept LOMENDOR signals as a function of 6 with S, = 0.55. Electron saturation factor: (A) S, = 0.33; (B) S, = 3. Calculated from set [5] (solid lines); measured for the (Y protons in a MAU single crystal (triangles). Same arbitrary units as those in Fig. 3.

sions [ 121, [ 131. As a matter of fact the experimental curve at S, = 0.02, S, = 0.08 presents a full width at half-maximum equal to 12.7 kHz very close to the value 12.2 1 kHz, calculated from Eq. [ 131.

When S, increases, the lineshape exhibits broadening. A further increase in the nuclear saturation factor gives rise to some oscillations. When the electron saturation factor is also increased, an absolute maximum appears in the lineshape (Fig. 4) whose 6 value differs from zero and increases with increasing S, and S, values.

The second harmonic LOMENDOR signal amplitude shows a dependence on frequency difference and saturation factors quite different from that of the first harmonic one. The linewidth of LZ, according to Eq. [ 141, is much smaller than that of LI . This is shown in Fig. 4 where we report both theoretical (solid lines) and experimental (triangles) results for the first and second harmonic spectra on the same scale for the case of S, = 0.55 and different electron saturation factor values: S, = 0.34, s, = 3.

Up to this point we have taken into consideration RF-swept spectra, obtained at different 6 values. It is also possible to record variable frequency LOMENDOR spectra by sweeping the frequency difference 6. An example of this method is shown in Fig. 5 where the RF-swept spectrum at 6 = 5 kHz is compared with the d-swept spectrum. From the figure it is seen that the LOMENDOR signal linewidth depends on T,,, while the VF-LOMENDOR signals exhibit a narrow line whose half-width at half-maximum depends on T,,. Also in this case the theoretical lineshapes confirm the validity of the experimental observations.


FIG. 5. First harmonic LOMENDOR spectra for the 01 protons in a MAU single crystal at same electron and nuclear saturation factor values S, = 0.12, S, = 0.12; (A) radiofrequency-swept case at 6 = 5 kHz, (B) &swept case. Solid lines: experimental lineshape; dotted lines: numerical solution of set [5].

Interesting features are found when the saturation curves at different 6 values are studied. Examples of these results are shown in Fig. 6 and Fig. 7 where the experimental values (triangles) of the first and second harmonic amplitudes at four values of 6 are reported as a function of the nuclear and electron saturation factors, respectively, in a semilogarithmic scale. The saturation effects and their dependence on 6 values clearly show up from the figures. In particular, as can be seen in Fig. 6, at a fixed 6 value and far from nuclear saturation, the first and second harmonics are proportional to S, and Si, respectively (I). By increasing S, this proportionality is gradually lost. The range of the nuclear saturation factor values in which such behavior holds changes with 6. Moreover the signal amplitude of the two harmonics attains its maximum at different nuclear saturation factor values. The position of these maxima shifts toward higher S,, values when 6 increases.

On the contrary, from the saturation curves for the first and second harmonic signals as a function of the electron saturation factor (Fig. 7), we observe that the saturation effects for both harmonics show up in the same range of S, values at every frequency explored. Furthermore, the trend of the second harmonic signal is considera- bly different. In fact a maximum of the signal is reached around 6 = 5 kHz and a further increase of 6 produces a signal reduction.

As described by Eqs. [8] and [ 141, Fig. 7 also shows that the harmonics have the same dependence on the m icrowave irradiation level. Far from electron saturation, the harmonic signals are always proportional to the third power of the m icrowave magnetic field, but, on increase of the electron saturation factor, the S, ranges for which this still holds change with the frequency difference 6 and with the harmonic


FIG. 6. Saturation curves for the (a) first and (b) second harmonic LOMENDOR signals versus S, and S, = 3. Calculated from set [5] (solid lines); measured for the (Y protons in a MAU single crystal (triangles) at four different 6 values: (A) 6 = 0.25 kHz; (B) 6 = 5 kHz; (C) d = 10 kHz; (D) d = 25 kHz. All quantities are in the same arbitrary units on a semilogarithmic scale.

considered. The numerical calculations (solid line) are a good confirmation of all such experimental results.

We have already demonstrated in Ref. (I) that, when the electron and nuclear saturation effects are negligible and I6T,, 1, ] 6T,, 1, ] 6T2, I G 1, the ratio I L2 I / IL, I between the second and the first harmonic LOMENDOR signals is equal to the S, value. If the latter condition is not satisfied, the ratio is proportional to S,,. This situa- tion is described well by Eq. [ 141 in which the dependence of the proportionality coefficient on the relaxation time T,p and T,, and on the 6 values is evident. In confirmation of these remarks, we report in Fig. 8 the experimental (triangles), analytical (dots), and theoretical (solid line) results for the ratio at S, = 0.12 for three different values of nuclear saturation factors as a function of 6. Since Eq. [ 151 is valid far from nuclear saturation, the analytical results, given by Eq. [ 141, are compared with experimental and numerical results only for the case S,, = 0.12.

Finally, note that, by the use of Eq. [ 151, the determination of the longitudinal nuclear relaxation time Tlnf is possible without a knowledge of the RF field inten- sities by three measurements of the ratio between the second and the first harmonic signals.

CONCLUSIONS

In this paper, we have proposed an extension of the LOMENDOR technique, which, together with a high degree of experimental simplicity, presents considerable


lo- 1 10 Se

FIG. 7. Saturation curves for the (a) first and (b) second harmonic LOMENDOR signals versus S, with S, = 0.83. Solid lines: numerical solution of equation set [5]. Triangles: experimental results for the (Y protons in a MAU single crystal at the same d values as those in Fig. 6.

10-l

10-2

10-3

FIG. 8. Ratio between the second and the first harmonic LOMENDOR signals as a function of 6 with S, = 0.12. Nuclear saturation factor: (a) .S, = 0.12; (b) S, = 0.6; (c) S, = 3. Solid lines correspond to the numerical solution of set [S], dotted lines to the approximate Eq. [ 151. Triangles give experimental results for the rr protons in a MAU single crystal.


advantages to study electron and nuclear relaxation processes. In fact, it has been found that this new variable frequency experimental procedure can yield a narrow line whose position is independent of the static magnetic field. The width of this line, if TI, ti T2,,, depends on the value of TIP only; therefore a direct and immediate measure ofthis relaxation time can be obtained. Moreover, since it often happens that I+‘, B W, (this is the case in organic radicals), an easy determination of the electron relaxation transition probability W, is allowed by Eq. [6c].

We have also shown the VF-LOMENDOR lineshape characteristics and its dependence on the microwave and radiofrequency irradiation level.

Finally, the availability of data relative to two different harmonics of the electron transverse magnetization components allows us to obtain direct information on the nuclear longitudinal relaxation time.

ACKNOWLEDGMENTS

The author thanks Dr. R. Ambrosetti for his useful reading of the manuscript. He is very grateful to Mr. P. Palla and to Mr. R. Michelotti for careful help given in drawing the figures and in setting up the electronic experimental apparatus, respectively. Furthermore, he is indebted to Mr. A. Giannotti for technical support.

REFERENCES

1. C. PINZINO, J. Magn. Reson. 81,318 (1989). 2. A. DI GIACOMO AND S. SANTUCCI, Nuovo Cimento B 63,407 ( 1969). 3. M. MARTINELLI, L. PARDI, C. PINZINO, AND S. SANTUCCI, Phys. Rev. B 16, 164 (I 977). 4. D. H. WHIFFEN, Mol. Whys. 10,595 (1966). 5. L. KEVAN AND P. A. NARAYANA, “Multiple Electron Resonance Spectroscopy” (M. M. Dorio and

J. H. Freed, Eds.), Plenum, New York, 1979. 6. M. GIORDANO, M. MARTINELLI, L. PARDI, S. SANTUCCI, ANDC. UMETON, J. Mugn. Reson. 64,47

(1985). 7. A. COLLIGIANI, C. PINZINO, AND M. BERTOLINI, J. Mugn. Reson. 33,5 111 (1979). 8. A. COLLIGIANI, C. PINZINO, M. BRUSTOLON, C. CORVAJA. G. BANDOLI, AND D. A. CL.EMENTI, Mol.

Phys. 39, 1 I53 ( 1980).

longitudinally modulated endor spectroscopy. ii. variable frequency measurements

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