Evaluation of flash CIDNP experiments by iterative reconvolution

Volume 165. number 1

CHEMICAL PHYSICS LETTERS

5 January 1990

EVALUATION OF FLASH CIDNP EXPERIMENTS BY ITERATIVE RECONVOLUTION Martin GOEZ Institutfiir Physikalische und Theoretische Chemie, Technische UnversitiitBraunschweig, HansSommer-Strusse 10, D-3300 Braunschweig, Federal Republic oiGermany Received 22 August 1989; in final form 3 October I989

The time resolution of flash CIDNP experiments can be increased by the use of staggered observation pulses, and iterative reconvolution of pulse and CIDNP magnetization. The general solution for the convolution of an arbitrary rfpulse shape with a time-dependent nuclear spin polarization is derived. It is shown that correct rate constants are obtained only if the true pulse shape is used in the evaluation. A measurement of the photoreaction between anthraquinone and N,Ndimethylaniline is presented as an experimental test.

1. Introduction

Chemically induced dynamic nuclear polarization (CIDNP) denotes the transient occurrence of anomalous NMR line intensities from the products of a chemical reaction carried out in a magnetic field (e.g. of an NMR spectrometer) [ 11. The phenomenon is caused by a spin sorting mechanism taking place in an intermediate radical pair. The polarization evolves during the lifetime of this pair (w 10 ns), but persists in the products for the spin-lattice relaxation time T,. Thus this effect is well suited for the study of fast radical reactions in solution, in spite of the inherently slow timescale of NMR measurements. Most CIDNP investigations of photoreactions have been performed using steady-state illumination. Valuable information about the mechanism of a reaction, but little about is kinetics can be gained in this way. However, with pulsed FT-NMR spectrometers it is equally feasible to perform time-resolved CIDNP measurements (“flash CIDNP”) [2-8 1. In these experiments, the chemical reaction is started by a flash, usually from a pulsed laser. The nuclear spin states of the products are probed shortly afterwards with a radiofrequency (rf ) pulse. A kinetic profile is obtained by variation of the delay between flash and observation pulse. Since the generation of nanosecond laser pulses presents no problem, the width of the probing rfpulse

(microsecond) determines the time resolution of the CIDNP detection method. Iterative reconvolution can be used to increase this time resolution considerably. A mathematical treatment of the convolution of rfpulse and chemical kinetics is given in this paper.

2. Results and discussion Ideally, the magnetization caused by the CIDNP effect should change only slowly compared to the duration of the probing rf pulse. In practice, this condition is not met. The minimum usable pulse width for our NMR spectrometer (Bruker WM-250) is 0.5 ps for protons (corresponding to a flip angle of about 7” ). Thus, even for very low reactant concentrations (= 10e4 M) the time constant of a diffusion controlled pseudo-first-order reaction in a non-viscous solvent (e.g. acetonitrile) is comparable to the pulse length. A booster amplifier has been used to shorten the observation pulse while maintaining the flip angle [ 9 1, but this demands among other things a redesign of probeheads, since commercial ones do not have the necessary voltage rating. Besides, for a given flip angle the necessary rf power increases with the inverse square of the pulse width, thus soon becoming impractically high. The other approach is to stagger the observation

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pulses between experiments by a time interval d

smaller than their duration T [6] #i. Their finite width is taken into account mathematically, as described below. In this way, a time resolution equal to the stagger interval (typically 50- 100 ns) can be obtained without modification of the NMR spectrometer. The cost of this is a reduction of the signal to noise ratio by a factor of approximately T/A. For one special case (rectangularly shaped pulses), the result for the convolution of rf pulse and CIDNP magnetization has been given in ref. [ 61. In the following, the general solution for arbitrary pulse shapes will be derived. It is sufficient to analyze the situation classically in a Cartesian frame rotating with the Larmor frequency of the nucleus in question. The observation pulse of magnetic field strength o (in frequency units) is static in this frame and taken along the xaxis. In the general case, the z-magnetization is a superposition of a constant background magnetization M,, and a changing term Mkln(t) due to the CIDNP effect. Application of the rf pulse causes a rotation of the magnetization vector around the x-axis, as described by the following system of differential equations:

mediately after the pulse we get the two solutions M,,,, = const. exp [ ? ip( 1‘) ] .

(5)

With this the inhomogeneous system of differential equations can be solved in a straightforward way, using the method of variation of constants. The general solution MX._, of ( 1)- ( 3 ) for a pulse starting at T, is given by

(6)

M,=MXO >

M,=M,cos[(D(T)]tM~sin[yl(T)] T + M,,,(t+To)

s

a(t) cos[ul(T)-p(t)

1dt,

(7)

dt,

(8)

0

M,=M,cos[yl(T)]-Mtisin[q(T)] -

lMkln(f+To) Nt)sin[dT)--dt)l

where M,, MN, and Mro are the components of a possible background magnetization. For the special case of a rectangular pulse, and in the absence of transverse background magnetization, eq. (7) yields

T

+W

s

Mkin(t+ To) COS[W(T-t)]

dt ,

(9)

0

or, alternatively (compare ref. [ 6 ] ) M,,=Md

Without the perturbation term dMkin(t)/dt, the remaining homogeneous system of differential equations simply describes a rotation of a constant magnetization vector M with changing angular velocity o( t ), The result is a rotation through an angle p( T),

(4)

where the integration has to be taken over the actual length T of the rf pulse. For the y-magnetization im-

12

sin(wT)

T

dh’~~,(t+To)sin[w(T-t)]

dt.

(10)

In most flash CIDNP experiments, Mti and My0 are zero, and Mfi is removed by presaturation [ lo]. Also, normally small flip angles are used, so that the cosine term in the integral of (7) may be taken as equal to one. Under these assumptions, eq. (7) is reduced to T My=

” Note that eqs. (50 and (7) as well as (8) in ref. [ 6 ] are partially incorrect and should be replaced by eqs. (lo), and (9) of the present work.

sin(oT)+&.(T6)

s

Mkin(t+To)W(t)

dt.

(11)

0

If

no

closed form solution exists for Mkin( t), or if

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its time derivative is more convenient to handle, it may be advantageous to convert ( 11) by integration by parts into

+

$lM,.(I+ To) dt * )

[v(T)-T(~) 1

(12)

Frequently, Mkin(t) contains exponential terms exp( -t/r). For these, a decomposition of the righthand side of ( 11) is possible M,,=exp(-To/r)

I

exp(-f/r)w(l)dt.

(13)

0

The integral in this equation is now independent of To. Thus, observation of an exponential signal term with staggered pulses just amounts to multiplying it with a constant scaling factor. For physically reasonable pulse shapes, this factor decreases monotonically with the ratio r/T. Eqs. ( 1l)-( 13) allow the calculation of theoretical signal intensities for a given reaction mechanism, if o is known. Rate constants can be obtained by performing this convolution iteratively, until x2 between experiment and model is minimized. As an example, the photoreaction between anthraquinone and N,N-dimethylaniline was measured in acetonitrile-&. The basic experimental setup is described in ref. [ 111. The delay between laser flash and rf pulse was determined with a precision of 2.5 ns with a fast photodiode. An excellent signal to noise ratio was reached by careful optimization of experimental parameters [ I2 1. The envelope of the pulse was measured by coupling out approximately 1% of it to an oscilloscope (Tektronix 475, rise time 1.75 ns) with a reflectometer (Rohde & Schwarz, ZDP-BN 35691). In this configuration, the influence of the measuring device on the impedance matched network of rf amplifier and probehead is negligible. Convolution with exp( -o,t/2Q), the envelope of the impulse response of a series LRC-circuit (resonance angular frequency rue, quality factor Q) yields the envelope of the magnetic field of the rf pulse. For our apparatus, pulses were found to be characterized by exponentially rising and decaying flanks with nearly the

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same time constant ( z 160 ns). Thus, a pulse of 0.5 ys nominal duration is significantly longer, and its center of gravity is shifted towards later times. The mechanism of the investigated reaction is known [ 6,13 1. A radical pair is formed in a diffusion-controlled electron transfer reaction between N,N-dimethylaniline and electronically excited anthraquinone in its triplet state. In-cage back electron transfer regenerates the starting materials and leads to an emission NMR signal for the dimethylamino protons of the aniline molecules produced in this way, in accord with Kaptein’s rules. Escaping aniline cations are converted to neutral molecules by degenerate electron exchange with their parent compounds. Since only a negligible part of the spin polarization is lost by relaxation in the free radical cations [ 121, a polarization of almost equal magnitude, but opposite sign results for the methyl protons of the aniline molecules formed in this reaction channel. Reactant concentrations were chosen such that all bimolecular steps in the kinetic scheme can be treated as pseudo-first-order reactions. In this case, a closed form solution of the rate equations exists [ 141. The time dependence of the polarization is given by a superposition of a negative fast rising exponential term, another exponential of opposite sign with smaller time constant, and a negligibly small negative dc term. Fig. 1 displays the results of fitting a biexponential rate law with the convolution ( 11) to the experimental data. For this, a Marquardt algorithm was adapted from t 141. Using the decomposition ( 13) this algorithm was modified such that integrals have to be calculated only once for each set of rate constants. Significantly lower run times of the fit program result for larger data sets. This is especially important for complicated pulse shapes or large flip angles where integration has to be performed numerically. Fits are shown for a hypothetical rectangular pulse envelope, as well as the measured one. The significant differences both in the obtained time constants, and in the tit quality in these two cases prove that the real pulse shape has to be taken into account. From the measurement, the rate constant for the degenerate electron exchange between N,N-dimethylaniline and its radical cation at 267 K was de13

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CHEMICAL PHYSICS LETTERS

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Acknowledgement This work was supported by the Deutsche Forschungsgemeinschaft. Thanks to S. Breide for providing the reflectometer, and to Professor Dr.H. Dreeskamp for helpful discussions.

References

80

70

6C4

0

,‘I

1

200

LOO 600

9’

I’I’I’I’I

800

1000 1200 X00

.!i ”

Fig. 1. Evaluation of a measurement of the photoreaction of 8.0~ 10m4M anthraquinone with 3.2~ 10T4M N,N-dimethylaniline in acetonitrile-dX (T= 267 K, L,,,=343 nm, width of rf pulse 0.5 ps, stagger interval 100 ns, 16 x 256 scans) by iterative reconvolution. Normalized power spectra integrals of the dimethylamino proton signals at different delays lo between laser discharge and rf pulse are depicted as bullets, lines denote the best lit resulting from two different pulse shapes. Due to the power spectrum representation, the signal phases appear inverted.

termined to be 1.3 x lo9 M-’ s-‘. A full investigation of the kinetics and mechanism of this reaction based on the described technique will be published separately [ 121.

14

[ 11 C. Richard and P. Granger, Chemically induced dynamic nuclear and electron polarization - CIDNP and CIDEP (Springer, Berlin, 1974); L.T. Muus, P.W. Atkins, K.A. McLauchlan and J.B. Pedersen, eds., Chemically induced magnetic polarization (Reidel, Dordrecht, 1977); KM. Salikhov, Yu.N. Molin, R.Z. Sagdeev and A.L. Buchachenko, eds., Spin polarization and magnetix effects in radical reactions (Elsevier, Amsterdam, 1984), and references therein. [ 21 G.L. Gloss, R.J. Miller and O.D. Redwine, Accounts Chem. Res. 18 ( 1985) 196, and references therein. [3]G.L. Gloss and O.D. Redwine, J. Am. Chem. Sot. 107 (1985) 4543. [ 41 J.K. Vollenweider, H. Fischer, J. Hennig and R. Leuschner, Chem.Phys.97 (1985) 217. [ 51 R. Leuschner and H. Fischer, Chem. Phys. Letters 12 I (1985) 554. [ 61 M. Laufer, Chem. Phys. Letters 127 ( 1986) 136. [ 71 J.K. Vollenweider and H. Fischer, Chem. Phys. 108 (1986) 365. [ 81 R. Hany, J.K. Vollenweider and H. Fischer, Chem. Phys. 120 (1988) 169. [ 91 R.J. Miller and G.L. Gloss, Rev. Sci. lnstr. 52 ( 1981) 1876. [ IO] S. Schlublin, A. Wokaun and R.R. Ernst, J. Magn. Reson 27 ( 1977) 273. 1I ] M. Llufer and H. Dreeskamp, J. Magn. Reson. 60 ( 1984) 357. 121 M. Goez, in preparation. 13) D. Roth, in: Chemically induced magnetic polarization, eds. L.T. Muus, P.W. Atkins, K.A. McLauchlan and J.B. Pedersen (Reidel, Dordrecht, 1977) p. 39. [ 141 W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterhng, Numerical recipes (Cambridge Univ. Press, Cambridge, 1986).