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Forbidden-transition-labelled EPR (FORTE)
18 November 1994
ELSEVIER
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 230 ( 1994) 67-74
Forbidden-transition-labelled EPR ( FORTE ) . An approach for the sensitive measurement of forbidden EPR transitions Michael Willer, Arthur Schweiger Laboratorium ftir Physikalische Chemie, Eidgenijssische Technische Hochschuie, CH-8092 Zurich, Switzerland
Received 1 August 1994; in final form 10 September 1994
Abstract A pulsed EPR method based on a hole-burning experiment is described which allows the sensitive measurement of forbidden EPR transitions in single crystals, powders and frozen solutions. With this approach the field positions of the forbidden transitions are labelled by strong peaks in a two-dimensional spectrum, well separated from the allowed transitions. The basic principles of the technique, model calculations and first experimental results are presented.
1. Introduction The frequencies of forbidden EPR transitions where the electron spin and one or several nuclear spins flip simultaneously contain important information about an electron-nuclear spin system. However, in the standard field-swept EPR display the forbidden transitions are usually weak in intensity and are often obscured by the allowed transitions. Forbidden transitions may sometimes be observed in single crystal and powder samples if the anisotropic part of the hyperfine interaction and/or the nuclear quadrupole couplings are large and the individual EPR lines are sufficiently narrow. A number of techniques for extracting the peak positions of forbidden EPR lines have been proposed, including L-band parallel mode EPR [ 11, electron spin transient nutation experiments [ 21, and multiquantum EPR [ 3 1. In this Letter, we present a pulsed two-dimensional EPR method for measuring forbidden transitions with high sensitivity. The approach is based on the concept of FID-detected hole burning [4] and enables
one to completely unravel allowed and forbidden transitions. The pulse sequence for this experiment is shown in Fig. 1. A selective microwave (mw) pulse drives forbidden EPR transitions thereby burning a pattern of transient holes into the EPR spectrum, which is detected via the free induction decay (FID) following a non-selective x/2 pulse [ 51. The hole
n/2
-
w
preparation
detection
Fig. 1. Pulse sequence for the FORTE experiment. Typical parameters: length of the Gaussian-shaped selective preparation pulse t$t, = l-5 ps (fwhh); length of the non-selective 2/2 readout pulse t F$=’ = 1O-20 ns.
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)01133-g
68
h4. Wilier, A. Schweiger / Chemical Physics titters 230 (1994) 67- 74
pattern representing the nuclear transition frequencies is then used to label the position of the forbidden transitions in the B0 field swept EPR display. As a result of this forbidden-transition-labelled EPR (FORTE) experiment the peaks in a 2D plot, &, field versus sublevel frequency, represent the spectrum of the forbidden EPR transitions with intensities that are strongly enhanced and well separated from the allowed transitions.
2. Basic procedure For the description of the underlying principles of FORTE, it is sufficient to consider an anisotropic S= l/2, I= l/2 electron-nuclear spin system. With the nucleus lying in the xz plane of the laboratory frame, the rotating frame Hamiltonian in angular frequencies is given by ~=~~S,+o,I,+AS,I,+BS,I,,
where 2r] is the angle between the quantization axes of the nuclear spin in the two m, states. The inhomogeneity rinh of the EPR lines, caused by the superposition of many spin packets differing in their offset frequency J&, is assumed to be small compared to the hyperfine coupling. We first consider the behavior of a particular spin packet during the preparation period. If the selective mw pulse is applied to one of the four electron spin transitions, the energy levels involved can be treated as a virtual two-level system. A selective pulse with mw frequency o,, on-resonant with the forbidden transition ( 1, 4) with frequency wf=c+4 (Fig. 2a), changes the population between the two levels involved. The population distribution after the pulse depends on the flip angle a)
(I),
where QS=ws-~mw=g,&ABO/h is the offset of the electron Zeeman frequency os from the mw fre-, is the nuclear Zeeman quency mrnwl ol= -g&,BJfi frequency, and A and Bare related to the elements A, of the hyperfine matrix by A = A, and B = A,. The frequencies of the two allowed (Ams= + 1, Aml=Q) and the two forbidden (Ams= k 1, Am,= k 1) electron transitions are given by [ 6 ] Cl
d)
o,=Q~~~(w,-qj)) and w+2,*&0,+wg))
(2)
l&?-yF2 j’...
with the two nuclear transition
frequencies ai
3 ,-’
The corresponding
EPR intensities
I, =cos+
>
and Ir = sin’rl =
Ic+b(%+cQ21
..;.:‘a
.:,a,: ..
../f .’ ,.,’ ,,.j ,,,’ 9
t
4
can be expressed
by I&-$(wc-qd’l % op
. . ..f
(4)
Fig. 2. Qualitative explanation of the FORTE experiment based on the model system S= l/2, I= l/2. (a) Population distribution for a particular spin packet after excitation of the forbidden transition ( 1,4) by a selective pulse with effective flip angle /3r= x. (b) EPR spectrum with hole pattern (top) and corresponding sublevel spectrum (bottom). (c ) Population distribution after excitation of the allowed transition (2,4) by a selective pulse with effective flip angle pB= (2n - 1 )n. (d) EPR spectrum with hole pattern (top) and corresponding sublevel spectrum (bottom).
69
M. Wilier, A. Schweiger / Chemical Physics Letters 230 (1994) 67- 74
where o,=g&B,/fi is the mw field strength in angular frequency units, t:L is the length of the selective mw pulse and j&=w, t& denotes the nominal flip angle (flip angle for a transition matrix element of unity). For a flip angle j?r= x full inversion of the on-resonant polarization is achieved. As a consequence, for this particular flip angle, the polarization of the two allowed transitions (1,3) and (2,4) becomes zero, whereas the polarization of the off-resonance forbidden transition (2,3) remains unchanged (Fig. 2a). Thus, the selective mw pulse burns a pattern of holes into the inhomogeneously broadened EPR lines. For the spin packet representing the centers of the inhomogeneous EPR lines, a hole of depth 21, is burned into the forbidden EPR line ( 1,4) (population inverted), and holes of depths Z, are burned into the allowed EPR lines ( 1,3) and (2,4) (populations equalized) (Fig. 2b, top). For a pulse length short compared to the phase memory time TM, the width of the holes (fwhh) is given by I’,,,,=5/t&. For tp> TM, the holes become shallower and rhole approaches the homogeneous linewidth r,,,, = 2/TM. The application of a short non-selective mw pulse of flip angle 7r/2 converts the whole polarization pattern into observable electron coherence that evolves during the detection period. If 1/rinh is small compared to the spectrometer deadtime rn, the transient signals stemming from the inhomogeneous lines decay within time rn. The FID then contains only the frequencies w,, og and zero. Since during detection under relaxation TM the shape of the holes is convoluted with the homogeneous line shape, the width of the peaks is broader than the width of the holes burned into the EPR lines during the preparation period. On a frequency axis with the excited forbidden transition frequency of= wi4 set to zero, the hole pattern represents the sublevel spectrum of the electronnuclear spin system, characterized by the nuclear transition frequencies W, and wg (Fig. 2b, bottom). For l/r,,, large compared to TD, as observed for example in single crystal samples, the contributions from the inhomogeneous lines can be eliminated by subtracting the FIDs of an experiment with and without the preparation pulse. Electron coherence cre-
ated by the selective pulse is eliminated by the twostepphasecycle, (O’,O’)-(O”, 180”). The crucial point of this experiment is to plot the hole (peak) pattern along an axis perpendicular to the frequency (BO field) axis representing the EPR spectrum (Fig. 3 ). The peaks on this w axis caused by the two deep holes burned into the strong allowed lines than label the position of the forbidden transition ( 1, 4). The forbidden transition is therefore indirectly detected via the allowed transitions having one energy level in common with the forbidden one. .4 new situation occurs, if the same experiment is carried out on an allowed transition (Figs. 2c and 2d). The effective flip angle of the (same) preparation pulse applied to the allowed transition (2,4) is given by LL=Po&.
(6)
For pr= K, we find Pa=Kmf.
(7)
Since for most experiments Z,/ZrB 1, preparation pulses with high nominal flip angles PO must be applied to fully invert the polarization of the forbidden transitions. As a consequence j3aB prso that the mag-
EPR (frquency umts)
-“P Fig. 3. Schematic contour plot consisting of two hemiplanes with an EPR axis and a sublevel frequency axis. The bent arrows mark the 90” rotations of the hole pattern along the o dimension around four rotation axes defined by the centers of the four EPR lines.
70
M. Wilier, A. Schweiger /Chemical Physics Letters 230 (1994) 6 7- 74
netization of an excited allowed transition will rapidly nutate in the plane perpendicular to the pumping field vector during the application of the preparation pulse. Since a rectangular preparation pulse with a large POgenerates holes with pronounced oscillatory tails, a Gaussian-shaped preparation pulse is used in this type of hole-burning experiment [ 4,5,79]. The quantity t&. then denotes the full width at half height of the pulse and PO is determined by the integral over the Gaussian pulse shape. With Gaussian pulses not only are the wiggles suppressed, but also the selectivity as compared to a rectangular pulse with the same area and the same width is increased [ 41. Depending on the ratio I,/I,, w1 and tgi the polarization of the allowed transition (2,4) after the pulse can have any value between the initial state (no hole) and full inversion (hole of depth 21,). In the first borderline case the spin system is again in thermal equilibrium after the pulse; holes are not created. In the second case shown in Fig. 2c, the polarization of the two forbidden transitions (2,3) and ( 1,4) is zero, and holes of depth Z, are burned into these lines (Fig. 2d, top). In both situations the polarization of the second allowed transition ( 1,3 ) remains unchanged (no hole). The hole spectrum is again plotted along an axis perpendicular to the one representing the EPR spectrum. Using quadrature detection, a peak is found at -w,, a second one at + wg (Fig. 3). Both peaks label the position of the allowed transition (2,4). The intensity of these peaks varies between zero (/3,= 2nx, n=l, ...) and Z&=(2n-l)rc, n=l, ...). and is therefore at least by a factor ZJZ, smaller than the peaks that label the forbidden transitions (Fig. 2d, bottom). In FORTE the intensity ratio between the peaks representing forbidden and allowed EPR transitions is thus at least increased by a factor > ( Z,/Zf) *. The two-dimensional plot of the B, field versus the Fourier transform of the FIDs, S( BO, w ), shows that the experiment is highly sensitive for measuring forbidden transitions across an EPR spectrum. In this display the field positions of the forbidden transitions are represented by the strong peaks appearing at w = ? w, and w = 2 wg well separated along the w dimension from the peaks of the allowed transitions at w= 0. The linewidth along the w axis is determined by the decay rate of the FID (in the optimum case by 2/T,), whereas in the B. field domain, the EPR lineshape is fully reproduced.
3. Numerical computations To give some insight into the behavior of multi-spin systems, we present a model calculation for a spin system with two I= l/2 nuclei. The numerical computations carried out with the simulation program GAMMA [ lo] consider all off-resonance effects caused by the non-ideality of the pulses in both the preparation and detection period. Electron coherence created by the selective pulse is eliminated with a two-step phase cycle, relaxation effects are neglected. The EPR spectrum of the three-spin system, S= 1/ 2, II = l/2, Z2= l/2, consists of four allowed transitions (Ams= ? 1, Aml, = 0, Am, = 0), eight singlespin-flip or singly forbidden transitions (four for nuclear spin I: Ams=fl, Aml,=?l, Amn=O (fl), and four for nuclear spin 2 : Ams= -t 1, Am,, = 0, Amrz = + 1 (f2) ) and four double-spin-flip or doubly forbidden transitions (Amp + 1, Am,, = f 1, Arnl1= + 1 (ff) ). The following parameters representing typical experimental situations are used: g = gisO= 2, mw frequency = 9 GHz, inhomogeneity ri,,/27C=2.35 MHz (Gaussian, fwhh), selective mw pulse: Gaussian with tf$ ~3.53 ys (fwhh); non-selective mw pulse; rectangular with tEF1 = 10 ns; nuclearspin 1:w1,/2~=-2MHz,A,/27c=10MHz,BL/ 2n: = 5 MHz; nuclear spin 2 : w,/2n= - 14 MHz, AJ 2x = 6 MHz, BJ27c = 3 MHz. These data result in the intensity ratios between forbidden and allowed transitions, Z,,/Z,=O.O316, ZQ/Z,=0.0126, Z,lZ,=O.O004. The EPR spectrum of this spin system is shown in Fig. 4a. Arrows mark two of the weak forbidden transitions of the type f2. Figs. 4b-4d show three contour plots obtained with this data set (magnitude spectra). Positive and negative frequency domains are the result of quadrature detection. In Fig. 4b the flip angle of the preparation pulse is optimum for the spin-flip transitions of nucleus 1 (pn =K, /$,=5.75~). Each of the four forbidden transitions is labelled by a pair of peaks at frequencies ? w,, and ? wg,. Since A1 > I2w,, I the two peaks appear in different hemiplanes separated by the hyperfine splitting a, of nucleus 1. Along the B,-,field axis, the peaks with the same nuclear transition frequency are separated by the hyperfine splitting a2 of nucleus 2. In Fig. 4c the flip angle of the preparation pulse is optimum for nucleus 2 (&=n, /$,=9.1 In), i.e. the
71
M. Wilier, A. Schweiger / Chemical Physics Letters 230 (1994) 67- 74
a)
t x 20 -A-
+
/I/\
MJ\I\ I
x 20
w
i20.5
A_
\
F
321.5
b)
Bo/mT
322.5
4
20
320.5
Bo/mT
322.5
Bo/mT
320.5
Bo/mT
322.5
Fig. 4. Numerical FORTE spectra of the model systems S= l/2, I, = l/2, I,= l/2. (a) EPR spectrum (parameter: see text). (b)-(d) FORTE experiments: flip angle fir, =II optimized for the spin flips of nucleus 1 (Do= 5.75x) (b), flip angle &=K optimized for the spin flips of nucleus 2 &=9.1 lx) (c), flip angleA=n optimized for the double-spin flips (/I,,= 51.2~) (d).
peaks at nuclear transition frequencies *ma2 and + wgz have reached their maximum intensity. Here, A2 < 12~0,~I and the peaks associated with the same forbidden transition therefore appear in the same hemiplane. Note that in the w domain the peaks at frequencies w,~ and wgk are separated by the hyperfine splitting of nucleus k. A section taken parallel to the B. field axis of the 2D spectrum at one of these nuclear transition frequencies contains half of the singly forbidden EPR transitions of this nucleus. At frequency CO,, for example, a subset of two peaks is observed for nucleus 1 split by the hypertine coupling of nucleus 2 (Fig. 4b). The sum of the sections taken at the frequencies SW,] and ?wgL in the 2D plot can therefore be regarded as the singly forbidden EPR spectrum of nucleus 1. For a nucleus k the number of single-spin-flip transitions is given by 4Zk multiplied by all possible nuclear spin combinations of the other nuclei. In a spin system consisting of n nuclei with arbitrary nuclear
quantum number Z, the number transitions iVris therefore
N,=
5
k
4Zkfi r#k
(21,+1)
. >
of single-spin-flip
(8)
Taking into account the quadrupole coupling of nuclei with I> l/2, one observes, for example, at the sublevel frequency c&k of nucleus k a subset of n:+k (2Z,+ 1) peaks. For magnetic equivalent nuclei, Zkand Z, in Eq. ( 8 ) must be replaced by the corresponding maximum group spin quantum numbers Fk and F,, respectively. With increasing flip angle double-spin-flip transitions (ff ) can also be excited. In Fig. 4d a flip angle of/3,=n (/3,=51.2x) is used. Since Zff~Zfl, IQ, the polarizations of the four single-spin-flip transitions after the pulse have a value between the initial state and inversion, consequently contributions of these single-spin flips are also observed in the 2D plot. The eight additional peaks stemming from double-spin
M. Wilier, A. Schweiger / Chemical Physics Letters 230 (1994) 67- 74
72
flips
occur
at
the
combination
frequencies,
den single-spin-flip transitions are observed by virtue of the small EPR lines generally found in this compound [ 111. The spin flips are caused by a simultaneous change in the spin state of the electron and one or several weakly coupled protons. Doublespin-flip transitions cannot be observed in this spectrum. With our experimental scheme however the positions of all single- and double-spin-flip transitions can be measured with high sensitivity and accuracy. Even indications of triple-spin flips have been found. The experiments have been carried out at 25 K with a home-built X-band pulsed EPR spectrometer [ 71. The EPR probehead contained a bridged loop-gap resonator with a quality factor of about Q=300 [ 12,131. The FIDs have been measured with a 500 MHz transient recorder, equipped with a fast adder for signal averaging (LeCroy model 7200 digital osPreparation pulses approximately cilloscope). Gaussian in shape have been used [ 71.
* (~ol~+wrwl), 2 (w~+c+i), * (w,~-m,i) and + ( osz-us1 ). The small artifacts in the 2D plot result from the fact that for high nominal flip angles even Gaussian pulses produce holes with some wiggles. They are caused by holes burned into the allowed transitions by off-resonance excitation, which are subsequently ‘transferred’ into the w domain during detection.
4. Experimental results To demonstrate the potential of the approach, we present some preliminary experimental results. The sample used in the experiment is a single crystal of (n-Bu,N)* [Ni ( mnt)2] (mnt = maleonitriledithiolate) doped with Cu (II) ions ( 63,65Cu: Ni = 1: 200). The cw EPR spectrum of the rnp = - 312 low-field transition of 63Cu and 65Cu recorded at room temperature is shown in Fig. Sa. In addition to the two allowed transitions weak satellite lines due to forbid-
4
b) /
cl 293
%U
Bo/mT
296
297
293
Bo/mT
296
291
5
WI \
I
10
15 N 3 G
i 25
30
Fig. 5. Experimental demonstration of the FORTE experiment on a Cu(II)-doped (n-BqN), [Ni(mnt),] single crystal. (a) Low-field part, rnp = -3/2, of the cw EPR spectrum measured at room temperature. Arrows indicate single-spin flips. (h),(c) Corresponding pulse contour plots, temperature 25 K, r”,“,“’ = 20 ns, tf,, = 2.28 us (fwhh of the Gaussian), with a nominal flip angle of the preparation @,=8x (b),/I,,c40x (c).
M. Wilier, A. Schweiger /Chemical Physics Letters 230 (1994) 67- 74
The length of the Gaussian-shaped preparation pulse was tS,et,= 2.28 us (fwhh), the length of the nonselective pulse was tg?’ =20 ns. The B0 field was swept over a range of 5.5 mT, incremented in steps of A&=0.05 mT. For every field position 200 FIDs with 200 data points each (time increment 10 ns) have been accumulated with a repetition rate of 1 kHz. The oscillations of the FID last up to 1.5 us. Electron coherence created by the selective preparation pulse has been eliminated by the two-step phase cycle mentioned above, and experiments without preparation pulse have been subtracted in order to eliminate contributions from the inhomogeneous line width. Experiments with different flip angles of the selective preparation pulse have been carried out. Fig. 5b shows the contour plot (magnitude spectrum) of a 2D experiment with a nominal flip angle POz 87~ For the two copper isotopes the proton single-spin-flip transitions in the w domain appear at the free proton frequency 12.5 MHz (w, =: wg% oI), with line intensities comparable to the allowed transitions observed at w z 0. The intensity of the allowed transitions at o, is below the detection limit. Increasing the strength of the preparation pulse (/$, z 40n) leads to the contour plot shown in Fig. 5c which contains additional peaks around 24.5 MHz (o,, + o,~ z upi + oB2 =: 2~~) representing double-spin-flip transitions (Am .=fl,Aml,=?l,Am,=?l),wheretwoproton spins and the electron spin flip simultaneously. Along the B, field axis, the two pairs of single-spinflip transitions of 63Cu and 65Cu are separated (in field units) by 0.86 mT (20,) each, whereas the two pairs of double-spin-flip transitions are separated by 40,. (Since for this particular orientation, the splitting of the two allowed EPR transitions of 63Cu and %u is close to 40,, two of the double-spin-flip transitions overlap. ) Peaks around 3 MHz in Figs. 5b and 5c are tentatively assigned to nitrogen atoms [ 141.
5. Concluding remarks FORTE is a powerful tool for the measurement of forbidden EPR transition frequencies, even in cases, where the weak forbidden lines are fully buried within the peaks of the allowed transitions. By a selective excitation of forbidden transitions during the prepa-
73
ration period, one obtains a ‘forbidden EPR spectrum’ well separated from the allowed one. The main advantage of the approach as compared to other experiments introduced for improving forbidden line intensities [ 1,2], is that the forbidden transitions in a spin system with small non-secular terms can be driven strongly and selectively for optimum signal intensity. Nevertheless two limitations must be considered. Electron spins are subject to TM relaxation during the preparation and detection period. Depending on this relaxation time one expects to find an optimum for t$.,, as far as depth (sensitivity) and broadening (selectivity) of the burned holes is concerned. A compromise must be made with respect to the pulse amplitude. On the one hand a high pulse power is needed for the strong driving of the forbidden transitions, but on the other hand an increase in pulse amplitude for reduces the selectivity a fixed time tf& correspondingly. A promising outlook is the application of FORTE to disordered systems such as powders and frozen solutions. In such samples the knowledge of forbidden transition frequencies would give access to magnetic parameters, in particular, the nuclear quadrupole interactions. Work in this direction is in progress.
Acknowledgement This research has been supported tional Science Foundation.
by the Swiss Na-
References [ 1] K.S. Rothenberger,
M.J. N&es, T.E. Altman, K. Glab, R.L. Belford, W. Froncisz and J.S. Hyde, Chem. Phys. Letters 124 (1986) 295. [ 21 A.V. Astashkin and A. Schweiger, Chem. Phys. Letters 174 (1990) 595. [ 3 ] H.S. Mchaourab, S. Pfenninger, W.E. Antholine, CC. Felix, J.S. Hydeand P.M.H. Kroneck, Biophys. J. 64 (1993) 1576. [4] Th. Wacker, G.A. Sierra and A. Schweiger, Isr. J. Chem. 32 ( 1992) 305. [ 5 ] Th. Wacker and A. Schweiger, Chem. Phys. Letters 186 (1991) 27. [ 61 A. Schweiger, in: Modern pulsed and continuous-wave electron spin resonance, eds. L. Kevan and M.K. Bowman (Wiley, New York, 1990).
74
M. Wilier, A. Schweiger / Chemical Physics Letters 230 (1994) 67- 74
[7] Th. Wacker, Ph.D. Thesis, ETH Ziirich, No. 9913, 1992.
[ 81 P. Schosseler, Th. Wacker and A. Schweiger, Chem. Phys. Letters 224 (1994) 319.
[ 91 C. Bauer, R. Freeman, T. Frenkiel, J. Keeler and A.J. Shaka, J. Magn. Reson. 68 ( 1984) 442. [ lo] S.A. Smith, T.O. Levante, B.H. Meier and R.R. Ernst, J. Magn. Reson. A 106 (1994) 75.
[ 111 J.S. Annie and P.T. Manoharan, Spectrochim. Acta 48 A (1992) 1715.
[ 121 S. Pfenninger, J. Forrer, A. Schweiger and Th. Weiland, Rev. Sci. Instr. 59 (1988) 752. [ 13 ] J. Forrer, S. Pfenninger, J. Eisenegger and A. Schweiger, Rev. Sci. Instr. 61 (1990) 3360. [ 141 E.J. Reijerse, A.H. Thiers, R. Kanters, M.C.M. Gribnau and C.P. Keijzers, Inorg. Chem. 26 (1987) 2764.
ELSEVIER
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 230 ( 1994) 67-74
Forbidden-transition-labelled EPR ( FORTE ) . An approach for the sensitive measurement of forbidden EPR transitions Michael Willer, Arthur Schweiger Laboratorium ftir Physikalische Chemie, Eidgenijssische Technische Hochschuie, CH-8092 Zurich, Switzerland
Received 1 August 1994; in final form 10 September 1994
Abstract A pulsed EPR method based on a hole-burning experiment is described which allows the sensitive measurement of forbidden EPR transitions in single crystals, powders and frozen solutions. With this approach the field positions of the forbidden transitions are labelled by strong peaks in a two-dimensional spectrum, well separated from the allowed transitions. The basic principles of the technique, model calculations and first experimental results are presented.
1. Introduction The frequencies of forbidden EPR transitions where the electron spin and one or several nuclear spins flip simultaneously contain important information about an electron-nuclear spin system. However, in the standard field-swept EPR display the forbidden transitions are usually weak in intensity and are often obscured by the allowed transitions. Forbidden transitions may sometimes be observed in single crystal and powder samples if the anisotropic part of the hyperfine interaction and/or the nuclear quadrupole couplings are large and the individual EPR lines are sufficiently narrow. A number of techniques for extracting the peak positions of forbidden EPR lines have been proposed, including L-band parallel mode EPR [ 11, electron spin transient nutation experiments [ 21, and multiquantum EPR [ 3 1. In this Letter, we present a pulsed two-dimensional EPR method for measuring forbidden transitions with high sensitivity. The approach is based on the concept of FID-detected hole burning [4] and enables
one to completely unravel allowed and forbidden transitions. The pulse sequence for this experiment is shown in Fig. 1. A selective microwave (mw) pulse drives forbidden EPR transitions thereby burning a pattern of transient holes into the EPR spectrum, which is detected via the free induction decay (FID) following a non-selective x/2 pulse [ 51. The hole
n/2
-
w
preparation
detection
Fig. 1. Pulse sequence for the FORTE experiment. Typical parameters: length of the Gaussian-shaped selective preparation pulse t$t, = l-5 ps (fwhh); length of the non-selective 2/2 readout pulse t F$=’ = 1O-20 ns.
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)01133-g
68
h4. Wilier, A. Schweiger / Chemical Physics titters 230 (1994) 67- 74
pattern representing the nuclear transition frequencies is then used to label the position of the forbidden transitions in the B0 field swept EPR display. As a result of this forbidden-transition-labelled EPR (FORTE) experiment the peaks in a 2D plot, &, field versus sublevel frequency, represent the spectrum of the forbidden EPR transitions with intensities that are strongly enhanced and well separated from the allowed transitions.
2. Basic procedure For the description of the underlying principles of FORTE, it is sufficient to consider an anisotropic S= l/2, I= l/2 electron-nuclear spin system. With the nucleus lying in the xz plane of the laboratory frame, the rotating frame Hamiltonian in angular frequencies is given by ~=~~S,+o,I,+AS,I,+BS,I,,
where 2r] is the angle between the quantization axes of the nuclear spin in the two m, states. The inhomogeneity rinh of the EPR lines, caused by the superposition of many spin packets differing in their offset frequency J&, is assumed to be small compared to the hyperfine coupling. We first consider the behavior of a particular spin packet during the preparation period. If the selective mw pulse is applied to one of the four electron spin transitions, the energy levels involved can be treated as a virtual two-level system. A selective pulse with mw frequency o,, on-resonant with the forbidden transition ( 1, 4) with frequency wf=c+4 (Fig. 2a), changes the population between the two levels involved. The population distribution after the pulse depends on the flip angle a)
(I),
where QS=ws-~mw=g,&ABO/h is the offset of the electron Zeeman frequency os from the mw fre-, is the nuclear Zeeman quency mrnwl ol= -g&,BJfi frequency, and A and Bare related to the elements A, of the hyperfine matrix by A = A, and B = A,. The frequencies of the two allowed (Ams= + 1, Aml=Q) and the two forbidden (Ams= k 1, Am,= k 1) electron transitions are given by [ 6 ] Cl
d)
o,=Q~~~(w,-qj)) and w+2,*&0,+wg))
(2)
l&?-yF2 j’...
with the two nuclear transition
frequencies ai
3 ,-’
The corresponding
EPR intensities
I, =cos+
>
and Ir = sin’rl =
Ic+b(%+cQ21
..;.:‘a
.:,a,: ..
../f .’ ,.,’ ,,.j ,,,’ 9
t
4
can be expressed
by I&-$(wc-qd’l % op
. . ..f
(4)
Fig. 2. Qualitative explanation of the FORTE experiment based on the model system S= l/2, I= l/2. (a) Population distribution for a particular spin packet after excitation of the forbidden transition ( 1,4) by a selective pulse with effective flip angle /3r= x. (b) EPR spectrum with hole pattern (top) and corresponding sublevel spectrum (bottom). (c ) Population distribution after excitation of the allowed transition (2,4) by a selective pulse with effective flip angle pB= (2n - 1 )n. (d) EPR spectrum with hole pattern (top) and corresponding sublevel spectrum (bottom).
69
M. Wilier, A. Schweiger / Chemical Physics Letters 230 (1994) 67- 74
where o,=g&B,/fi is the mw field strength in angular frequency units, t:L is the length of the selective mw pulse and j&=w, t& denotes the nominal flip angle (flip angle for a transition matrix element of unity). For a flip angle j?r= x full inversion of the on-resonant polarization is achieved. As a consequence, for this particular flip angle, the polarization of the two allowed transitions (1,3) and (2,4) becomes zero, whereas the polarization of the off-resonance forbidden transition (2,3) remains unchanged (Fig. 2a). Thus, the selective mw pulse burns a pattern of holes into the inhomogeneously broadened EPR lines. For the spin packet representing the centers of the inhomogeneous EPR lines, a hole of depth 21, is burned into the forbidden EPR line ( 1,4) (population inverted), and holes of depths Z, are burned into the allowed EPR lines ( 1,3) and (2,4) (populations equalized) (Fig. 2b, top). For a pulse length short compared to the phase memory time TM, the width of the holes (fwhh) is given by I’,,,,=5/t&. For tp> TM, the holes become shallower and rhole approaches the homogeneous linewidth r,,,, = 2/TM. The application of a short non-selective mw pulse of flip angle 7r/2 converts the whole polarization pattern into observable electron coherence that evolves during the detection period. If 1/rinh is small compared to the spectrometer deadtime rn, the transient signals stemming from the inhomogeneous lines decay within time rn. The FID then contains only the frequencies w,, og and zero. Since during detection under relaxation TM the shape of the holes is convoluted with the homogeneous line shape, the width of the peaks is broader than the width of the holes burned into the EPR lines during the preparation period. On a frequency axis with the excited forbidden transition frequency of= wi4 set to zero, the hole pattern represents the sublevel spectrum of the electronnuclear spin system, characterized by the nuclear transition frequencies W, and wg (Fig. 2b, bottom). For l/r,,, large compared to TD, as observed for example in single crystal samples, the contributions from the inhomogeneous lines can be eliminated by subtracting the FIDs of an experiment with and without the preparation pulse. Electron coherence cre-
ated by the selective pulse is eliminated by the twostepphasecycle, (O’,O’)-(O”, 180”). The crucial point of this experiment is to plot the hole (peak) pattern along an axis perpendicular to the frequency (BO field) axis representing the EPR spectrum (Fig. 3 ). The peaks on this w axis caused by the two deep holes burned into the strong allowed lines than label the position of the forbidden transition ( 1, 4). The forbidden transition is therefore indirectly detected via the allowed transitions having one energy level in common with the forbidden one. .4 new situation occurs, if the same experiment is carried out on an allowed transition (Figs. 2c and 2d). The effective flip angle of the (same) preparation pulse applied to the allowed transition (2,4) is given by LL=Po&.
(6)
For pr= K, we find Pa=Kmf.
(7)
Since for most experiments Z,/ZrB 1, preparation pulses with high nominal flip angles PO must be applied to fully invert the polarization of the forbidden transitions. As a consequence j3aB prso that the mag-
EPR (frquency umts)
-“P Fig. 3. Schematic contour plot consisting of two hemiplanes with an EPR axis and a sublevel frequency axis. The bent arrows mark the 90” rotations of the hole pattern along the o dimension around four rotation axes defined by the centers of the four EPR lines.
70
M. Wilier, A. Schweiger /Chemical Physics Letters 230 (1994) 6 7- 74
netization of an excited allowed transition will rapidly nutate in the plane perpendicular to the pumping field vector during the application of the preparation pulse. Since a rectangular preparation pulse with a large POgenerates holes with pronounced oscillatory tails, a Gaussian-shaped preparation pulse is used in this type of hole-burning experiment [ 4,5,79]. The quantity t&. then denotes the full width at half height of the pulse and PO is determined by the integral over the Gaussian pulse shape. With Gaussian pulses not only are the wiggles suppressed, but also the selectivity as compared to a rectangular pulse with the same area and the same width is increased [ 41. Depending on the ratio I,/I,, w1 and tgi the polarization of the allowed transition (2,4) after the pulse can have any value between the initial state (no hole) and full inversion (hole of depth 21,). In the first borderline case the spin system is again in thermal equilibrium after the pulse; holes are not created. In the second case shown in Fig. 2c, the polarization of the two forbidden transitions (2,3) and ( 1,4) is zero, and holes of depth Z, are burned into these lines (Fig. 2d, top). In both situations the polarization of the second allowed transition ( 1,3 ) remains unchanged (no hole). The hole spectrum is again plotted along an axis perpendicular to the one representing the EPR spectrum. Using quadrature detection, a peak is found at -w,, a second one at + wg (Fig. 3). Both peaks label the position of the allowed transition (2,4). The intensity of these peaks varies between zero (/3,= 2nx, n=l, ...) and Z&=(2n-l)rc, n=l, ...). and is therefore at least by a factor ZJZ, smaller than the peaks that label the forbidden transitions (Fig. 2d, bottom). In FORTE the intensity ratio between the peaks representing forbidden and allowed EPR transitions is thus at least increased by a factor > ( Z,/Zf) *. The two-dimensional plot of the B, field versus the Fourier transform of the FIDs, S( BO, w ), shows that the experiment is highly sensitive for measuring forbidden transitions across an EPR spectrum. In this display the field positions of the forbidden transitions are represented by the strong peaks appearing at w = ? w, and w = 2 wg well separated along the w dimension from the peaks of the allowed transitions at w= 0. The linewidth along the w axis is determined by the decay rate of the FID (in the optimum case by 2/T,), whereas in the B. field domain, the EPR lineshape is fully reproduced.
3. Numerical computations To give some insight into the behavior of multi-spin systems, we present a model calculation for a spin system with two I= l/2 nuclei. The numerical computations carried out with the simulation program GAMMA [ lo] consider all off-resonance effects caused by the non-ideality of the pulses in both the preparation and detection period. Electron coherence created by the selective pulse is eliminated with a two-step phase cycle, relaxation effects are neglected. The EPR spectrum of the three-spin system, S= 1/ 2, II = l/2, Z2= l/2, consists of four allowed transitions (Ams= ? 1, Aml, = 0, Am, = 0), eight singlespin-flip or singly forbidden transitions (four for nuclear spin I: Ams=fl, Aml,=?l, Amn=O (fl), and four for nuclear spin 2 : Ams= -t 1, Am,, = 0, Amrz = + 1 (f2) ) and four double-spin-flip or doubly forbidden transitions (Amp + 1, Am,, = f 1, Arnl1= + 1 (ff) ). The following parameters representing typical experimental situations are used: g = gisO= 2, mw frequency = 9 GHz, inhomogeneity ri,,/27C=2.35 MHz (Gaussian, fwhh), selective mw pulse: Gaussian with tf$ ~3.53 ys (fwhh); non-selective mw pulse; rectangular with tEF1 = 10 ns; nuclearspin 1:w1,/2~=-2MHz,A,/27c=10MHz,BL/ 2n: = 5 MHz; nuclear spin 2 : w,/2n= - 14 MHz, AJ 2x = 6 MHz, BJ27c = 3 MHz. These data result in the intensity ratios between forbidden and allowed transitions, Z,,/Z,=O.O316, ZQ/Z,=0.0126, Z,lZ,=O.O004. The EPR spectrum of this spin system is shown in Fig. 4a. Arrows mark two of the weak forbidden transitions of the type f2. Figs. 4b-4d show three contour plots obtained with this data set (magnitude spectra). Positive and negative frequency domains are the result of quadrature detection. In Fig. 4b the flip angle of the preparation pulse is optimum for the spin-flip transitions of nucleus 1 (pn =K, /$,=5.75~). Each of the four forbidden transitions is labelled by a pair of peaks at frequencies ? w,, and ? wg,. Since A1 > I2w,, I the two peaks appear in different hemiplanes separated by the hyperfine splitting a, of nucleus 1. Along the B,-,field axis, the peaks with the same nuclear transition frequency are separated by the hyperfine splitting a2 of nucleus 2. In Fig. 4c the flip angle of the preparation pulse is optimum for nucleus 2 (&=n, /$,=9.1 In), i.e. the
71
M. Wilier, A. Schweiger / Chemical Physics Letters 230 (1994) 67- 74
a)
t x 20 -A-
+
/I/\
MJ\I\ I
x 20
w
i20.5
A_
\
F
321.5
b)
Bo/mT
322.5
4
20
320.5
Bo/mT
322.5
Bo/mT
320.5
Bo/mT
322.5
Fig. 4. Numerical FORTE spectra of the model systems S= l/2, I, = l/2, I,= l/2. (a) EPR spectrum (parameter: see text). (b)-(d) FORTE experiments: flip angle fir, =II optimized for the spin flips of nucleus 1 (Do= 5.75x) (b), flip angle &=K optimized for the spin flips of nucleus 2 &=9.1 lx) (c), flip angleA=n optimized for the double-spin flips (/I,,= 51.2~) (d).
peaks at nuclear transition frequencies *ma2 and + wgz have reached their maximum intensity. Here, A2 < 12~0,~I and the peaks associated with the same forbidden transition therefore appear in the same hemiplane. Note that in the w domain the peaks at frequencies w,~ and wgk are separated by the hyperfine splitting of nucleus k. A section taken parallel to the B. field axis of the 2D spectrum at one of these nuclear transition frequencies contains half of the singly forbidden EPR transitions of this nucleus. At frequency CO,, for example, a subset of two peaks is observed for nucleus 1 split by the hypertine coupling of nucleus 2 (Fig. 4b). The sum of the sections taken at the frequencies SW,] and ?wgL in the 2D plot can therefore be regarded as the singly forbidden EPR spectrum of nucleus 1. For a nucleus k the number of single-spin-flip transitions is given by 4Zk multiplied by all possible nuclear spin combinations of the other nuclei. In a spin system consisting of n nuclei with arbitrary nuclear
quantum number Z, the number transitions iVris therefore
N,=
5
k
4Zkfi r#k
(21,+1)
. >
of single-spin-flip
(8)
Taking into account the quadrupole coupling of nuclei with I> l/2, one observes, for example, at the sublevel frequency c&k of nucleus k a subset of n:+k (2Z,+ 1) peaks. For magnetic equivalent nuclei, Zkand Z, in Eq. ( 8 ) must be replaced by the corresponding maximum group spin quantum numbers Fk and F,, respectively. With increasing flip angle double-spin-flip transitions (ff ) can also be excited. In Fig. 4d a flip angle of/3,=n (/3,=51.2x) is used. Since Zff~Zfl, IQ, the polarizations of the four single-spin-flip transitions after the pulse have a value between the initial state and inversion, consequently contributions of these single-spin flips are also observed in the 2D plot. The eight additional peaks stemming from double-spin
M. Wilier, A. Schweiger / Chemical Physics Letters 230 (1994) 67- 74
72
flips
occur
at
the
combination
frequencies,
den single-spin-flip transitions are observed by virtue of the small EPR lines generally found in this compound [ 111. The spin flips are caused by a simultaneous change in the spin state of the electron and one or several weakly coupled protons. Doublespin-flip transitions cannot be observed in this spectrum. With our experimental scheme however the positions of all single- and double-spin-flip transitions can be measured with high sensitivity and accuracy. Even indications of triple-spin flips have been found. The experiments have been carried out at 25 K with a home-built X-band pulsed EPR spectrometer [ 71. The EPR probehead contained a bridged loop-gap resonator with a quality factor of about Q=300 [ 12,131. The FIDs have been measured with a 500 MHz transient recorder, equipped with a fast adder for signal averaging (LeCroy model 7200 digital osPreparation pulses approximately cilloscope). Gaussian in shape have been used [ 71.
* (~ol~+wrwl), 2 (w~+c+i), * (w,~-m,i) and + ( osz-us1 ). The small artifacts in the 2D plot result from the fact that for high nominal flip angles even Gaussian pulses produce holes with some wiggles. They are caused by holes burned into the allowed transitions by off-resonance excitation, which are subsequently ‘transferred’ into the w domain during detection.
4. Experimental results To demonstrate the potential of the approach, we present some preliminary experimental results. The sample used in the experiment is a single crystal of (n-Bu,N)* [Ni ( mnt)2] (mnt = maleonitriledithiolate) doped with Cu (II) ions ( 63,65Cu: Ni = 1: 200). The cw EPR spectrum of the rnp = - 312 low-field transition of 63Cu and 65Cu recorded at room temperature is shown in Fig. Sa. In addition to the two allowed transitions weak satellite lines due to forbid-
4
b) /
cl 293
%U
Bo/mT
296
297
293
Bo/mT
296
291
5
WI \
I
10
15 N 3 G
i 25
30
Fig. 5. Experimental demonstration of the FORTE experiment on a Cu(II)-doped (n-BqN), [Ni(mnt),] single crystal. (a) Low-field part, rnp = -3/2, of the cw EPR spectrum measured at room temperature. Arrows indicate single-spin flips. (h),(c) Corresponding pulse contour plots, temperature 25 K, r”,“,“’ = 20 ns, tf,, = 2.28 us (fwhh of the Gaussian), with a nominal flip angle of the preparation @,=8x (b),/I,,c40x (c).
M. Wilier, A. Schweiger /Chemical Physics Letters 230 (1994) 67- 74
The length of the Gaussian-shaped preparation pulse was tS,et,= 2.28 us (fwhh), the length of the nonselective pulse was tg?’ =20 ns. The B0 field was swept over a range of 5.5 mT, incremented in steps of A&=0.05 mT. For every field position 200 FIDs with 200 data points each (time increment 10 ns) have been accumulated with a repetition rate of 1 kHz. The oscillations of the FID last up to 1.5 us. Electron coherence created by the selective preparation pulse has been eliminated by the two-step phase cycle mentioned above, and experiments without preparation pulse have been subtracted in order to eliminate contributions from the inhomogeneous line width. Experiments with different flip angles of the selective preparation pulse have been carried out. Fig. 5b shows the contour plot (magnitude spectrum) of a 2D experiment with a nominal flip angle POz 87~ For the two copper isotopes the proton single-spin-flip transitions in the w domain appear at the free proton frequency 12.5 MHz (w, =: wg% oI), with line intensities comparable to the allowed transitions observed at w z 0. The intensity of the allowed transitions at o, is below the detection limit. Increasing the strength of the preparation pulse (/$, z 40n) leads to the contour plot shown in Fig. 5c which contains additional peaks around 24.5 MHz (o,, + o,~ z upi + oB2 =: 2~~) representing double-spin-flip transitions (Am .=fl,Aml,=?l,Am,=?l),wheretwoproton spins and the electron spin flip simultaneously. Along the B, field axis, the two pairs of single-spinflip transitions of 63Cu and 65Cu are separated (in field units) by 0.86 mT (20,) each, whereas the two pairs of double-spin-flip transitions are separated by 40,. (Since for this particular orientation, the splitting of the two allowed EPR transitions of 63Cu and %u is close to 40,, two of the double-spin-flip transitions overlap. ) Peaks around 3 MHz in Figs. 5b and 5c are tentatively assigned to nitrogen atoms [ 141.
5. Concluding remarks FORTE is a powerful tool for the measurement of forbidden EPR transition frequencies, even in cases, where the weak forbidden lines are fully buried within the peaks of the allowed transitions. By a selective excitation of forbidden transitions during the prepa-
73
ration period, one obtains a ‘forbidden EPR spectrum’ well separated from the allowed one. The main advantage of the approach as compared to other experiments introduced for improving forbidden line intensities [ 1,2], is that the forbidden transitions in a spin system with small non-secular terms can be driven strongly and selectively for optimum signal intensity. Nevertheless two limitations must be considered. Electron spins are subject to TM relaxation during the preparation and detection period. Depending on this relaxation time one expects to find an optimum for t$.,, as far as depth (sensitivity) and broadening (selectivity) of the burned holes is concerned. A compromise must be made with respect to the pulse amplitude. On the one hand a high pulse power is needed for the strong driving of the forbidden transitions, but on the other hand an increase in pulse amplitude for reduces the selectivity a fixed time tf& correspondingly. A promising outlook is the application of FORTE to disordered systems such as powders and frozen solutions. In such samples the knowledge of forbidden transition frequencies would give access to magnetic parameters, in particular, the nuclear quadrupole interactions. Work in this direction is in progress.
Acknowledgement This research has been supported tional Science Foundation.
by the Swiss Na-
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