Wideband frequency-swept excitation in pulsed EPR spectroscopy

Highlights

• Offset-independent adiabaticity conveys flip angle and phase robustness.

• Some types of transverse interference can be refocused, but others cannot.

• Excitation bandwidth for pumped spins can exceed resonator bandwidth by far.

• Single-sideband upconversion reduces spurious excitation.

• Up to 800 MHz detection and at least 3 GHz excitation bandwidth are feasible.

Abstract

Excitation of electron spins with monochromatic rectangular pulses is limited to bandwidths that are smaller than the spectral widths of most organic radicals and much smaller than the spectral widths of transition and rare earth metal ions. With frequency-swept pulses, bandwidths of up to 800 MHz have previously been attained for excitation and detection of spin packets at frequencies of about 9.6 GHz and bandwidths of up to 2.5 GHz in a polarization transfer experiment at frequencies of about 34 GHz. The remaining limitations, mainly due to resonator bandwidth and due to pulse length restrictions are discussed. Flip angles for state-space rotations on passage of a transition can generally be computed from the critical adiabaticity by the Landau-Zener-Stückelberg-Majorana expression. For hyperbolic secant pulses, the Demkov-Kunike model describes excitation for spin packets within and outside the sweep range. Well within the sweep range, the Bloch-Siegert phase shift is proportional to critical adiabaticity to a very good approximation. Because of the dependence of both flip angle and coherence phase on critical adiabaticity, it is advantageous to use pairs of amplitude and frequency modulation functions that provide such offset-independent adiabaticity. Compensation for the resonator response function should restore offset-independent adiabaticity. Whereas resonance offsets and Bloch-Siegert phase can be refocused at certain pulse length ratios, phase dispersion in coupled spin systems cannot generally be refocused. Based on the bandwidth limitations that arise from spin dynamics, requirements are derived for a spectrometer that achieves precise spin control over wide bands. The design of such a spectrometer and hardware characterization by EPR experiments are discussed.

Introduction

The excitation bandwidth of monochromatic rectangular pulses in electron paramagnetic resonance (EPR) spectroscopy is of the order of 100 MHz. Very few paramagnetic species have spectra narrower than that. For example, spectral width for the widely used nitroxide spin labels ranges from about 180 MHz at X-band frequencies ($\sim 9.6\text{GHz}$) to about 450 MHz at W-band frequencies ($\sim 94\text{GHz}$). Spectra of triplet states of organic molecules, of transition metal complexes, and of rare earth ion complexes are usually wider than 1 GHz. Therefore, most experiments in pulse EPR spectroscopy are geared to the regime where only part of the spin packets of an inhomogeneously broadened EPR line is excited [1].

Such experiments have severe shortcomings. First, sensitivity may be lost by detecting the signal from only a fraction - sometimes a very small fraction - of all spin packets. In particular this applies to experiments where the excitation bandwidth could be increased without increasing the noise bandwidth of detection, such as electron spin echo envelope modulation (ESEEM) spectroscopy, its two-dimensional form of HYSCORE, electron-nuclear double resonance (ENDOR), and all pulsed electron-electron double resonance (ELDOR) experiments. Second, two transitions can be correlated in an experiment only if both of them are within the excitation bandwidth. Although ELDOR schemes enable correlation beyond the bandwidth of pulses at a single frequency, such approaches come at the expense of further sensitivity loss. This limitation also applies to ESEEM experiments, where the detected nuclear frequencies are differences of two transitions that must be excited in the same experiment. Third, spin packets at resonance offsets of the order of the excitation bandwidth of rectangular pulses follow spin dynamics that is not usually intended. The fraction of such spin packets is relatively large if the spectral line width is much larger than the excitation bandwidth. Therefore, spin control is much less precise in EPR spectroscopy than in NMR spectroscopy. As a result, each additional pulse causes sensitivity loss and often introduces coherence transfer pathways that lead to unwanted signal contributions. Such contributions cannot always be removed by phase cycling. Accordingly, multi-pulse techniques [2], which are prominent in NMR, are rarely used in EPR spectroscopy.

Related problems in broadband heteronuclear decoupling in liquid-state NMR, magnetic resonance imaging (MRI), and solid-state NMR of quadrupole nuclei have been addressed by application of frequency-swept pulses [3], [4], [5], [6], [7]. During the past decade or so, arbitrary waveform generators (AWGs) have become sufficiently fast to cover the full bandwidth of a few Gigahertz of microwave (MW) components in the pulse EPR spectrometers that are used for application work in the life sciences, in catalysis, and in materials research. Recently, the first commercial AWG setup with 1.6 GSa/s clock rate has been introduced by Bruker. Via Bruker’s intermediate MW frequency concept, this setup allows for EPR experiments at S band ($\sim 4\text{GHz}$), X band, Q band ($\sim 34\text{GHz}$), W band, and in the 263 GHz mm band. During the past few years the new opportunities due to shaped-pulse excitation with AWGs have been explored mainly by groups who worked with home-built setups [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. This work has revealed a great potential for frequency-swept excitation in pulsed EPR applications, but also some limitations that need to be considered in pulse sequence and spectrometer design. Here we present a critical review of these results and fill some gaps in previous description of spin dynamics during frequency-swept pulses.

The limitations of excitation by shaped pulses depend strongly on the type of spin system. Complications arise in multi-level systems and, in particular, for distributions of spin Hamiltonian parameters. Unfortunately, both these complications are typical for pulsed EPR applications. Many of these applications require separation of hyperfine or electron-electron dipole-dipole couplings from other interactions in macroscopically disordered systems. In our discussions, we always have such applications in mind.

This perspective article is structured as follows. First we consider passage of a two-level system, i.e., excitation of a single transition during a frequency sweep. We discuss the concept of critical adiabaticity $Q_{\text{crit}}$, the dependence of an equivalent flip angle on $Q_{\text{crit}}$ for linearly frequency-swept (chirp, WURST) and hyperbolic secant (HS) pulses, the compensation of the resonator response function that is necessary for attaining wideband uniform excitation of spin packets, and Bloch-Siegert phase shifts that arise during frequency-swept pulses. We then turn to passage of multi-level systems. In such systems, longitudinal and transverse interference effects arise in passage of several transitions that are connected by shared energy levels. The three-level system is treated separately, as it still allows for an analytical description of the outcome of a linear or HS frequency sweep. We further discuss the ladder topology, as it is encountered for electron group spins $S>1/2$, and systems where several energy levels are connected by a loop of allowed transitions.

Since most pulsed EPR experiments require echo formation, we then turn to refocusing of the phase dispersion induced by a frequency sweep and of the phase dispersion caused by the Bloch-Siegert effect. In multi-level systems, additional phase dispersion may arise from couplings and we discuss under which conditions such phase dispersion can be refocused.

We then consider additional limitations that are not caused by spin dynamics, but are rather imposed by instrumentation. First, we inquire how well hardware must be characterized in order to achieve reasonably precise spin control within the spin dynamics limits and how spurious excitation frequencies can be avoided. We then discuss what detection bandwidth can be achieved at the current level of spectrometer technology and point out that unobserved spins can be excited over a much wider band. From these considerations we derive a spectrometer design specification. We conclude with a short assessment of the new possibilities and open questions.