In 1963, Klein and Barton formulated the basic principles of the signal-to-noise ratio (SNR) when using rapid scans and spectral averaging [1
]. To paraphrase this work: if noise is white and two spectra are compared, the first acquired in a single scan in time T
with an integrating time constant τ and the second acquired by summing n
spectra, each acquired in time T/n
with integrating time constant τ/n
, the SNRs will be the same. However, if the noise has a 1/f
character, the latter method will exhibit lower noise. These authors present data in which multiple EPR spectra using magnetic field modulation and lock-in detection were acquired and averaged. In the present paper, we extend the method of Klein and Barton by using direct detection and compare results with data obtained using the conventional 100 kHz magnetic field modulation method, which also overcomes 1/f
noise. The applications are to spin labels, but the methodology and conclusions are widely applicable.
The method presented here utilizes non-adiabatic rapid-sweep (NARS) EPR. A small magnetic field sweep coil produces a triangular variation of the polarizing magnetic field over a spectral fragment at a repetition rate that is sufficiently high that 1/f noise arising from the receiver, the microwave source, and the environment is overcome. An analog-to-digital converter (ADC) with a high sampling rate converts the detected signal to a digital format, which is then averaged as the spectral fragment is repetitively swept. A series of spectral segments is collected across the spectrum and combined to arrive at a complete pure absorption spectrum. This spectrum will contain high frequency noise since it was acquired at high bandwidth. A digital filter is applied to remove this noise, which reduces the bandwidth in a manner that is analogous to use of an integrating time constant.
Bloch found two solutions to the phenomenological equations that bear his name, depending on whether or not the sweep of the magnetic field was adiabatic (i.e.
, so rapid that no thermal relaxation of excited spins occurs) or non-adiabatic such that the spin system remains in thermal equilibrium during sweep through the resonance condition [2
]. The adiabatic condition is given in Eq. (1)
in a form provided by Pake, where H1
is the magnitude of the incident RF field, Ho
is the static magnetic field, ω is the RF frequency, γ is the gyromagnetic ratio, and T1
is the spin-lattice relaxation time [3
The basic idea of NARS is to satisfy the condition for non-adiabatic sweep, Eq. (2)
, but at the same time to use a sufficiently rapid sweep repetition rate that 1/f
noise is avoided. Whether or not these conditions can be satisfied depends on the spin-lattice relaxation time.
A loop-gap resonator (LGR) is advantageous for NARS EPR because it exhibits a lower quality factor, Q
, and a higher filling factor, η, than a cavity resonator [4
]. The power needed to achieve the same H1
field at the sample is reduced, and the effects of phase noise and microphonics are diminished. The incorporation of a low-noise microwave amplifier (LNA) improves receiver performance. The amplifier is placed after the resonator, but before the signal mixer, to boost noise on the microwave carrier above the 1/f
noise of the receiver. Improvement in the performance of ADCs permits sampling frequencies that are fast relative to the polarizing field sweep. In this manner, sensitivity is greatly improved, even though the rate of sweep of magnetic field through resonance is reduced in accordance with Eq. (2)
. In summary, following Klein and Barton, the SNR is improved by increase in the number of accumulations per second, which diminishes the impact of 1/f
phase noise [1
We turn to a consideration of magnetic field modulation, which consists of superimposing a sinusoidal field of the form Hm
)] onto the static magnetic field, where Hm
is the modulation amplitude and ωm
is the modulation frequency. When resonance occurs, microwave sidebands arise ± n
away from the microwave carrier frequency. By offsetting the EPR microwave signal sidebands from the carrier and detecting them with a lock-in detector followed by an integrating time constant, microwave source phase noise is diminished, and if ωm
is sufficiently high, 1/f
detector noise and microphonics are also reduced [5
The noise in the system is ultimately determined by the integrating time constant, which establishes the receiver bandwidth. The digital filter in NARS detection and the use of the integrating time constant in the lock-in detection method are approximately equivalent. However, the lock-in method has a disadvantage because all magnetic field modulation causes spectral broadening and sacrifices EPR signal intensity. The extent of these effects is related to the modulation amplitude compared to the EPR linewidth. displays the effects of magnetic field modulation on a nitroxide undergoing intermediate motion using EasySpin, a MATLAB® (MathWorks™, Natick, MA) toolbox used to perform spectral simulations [6
], and pseudomodulation, a method that convolves sinusoidal modulation of an independent parameter of an arbitrary function with that function. It accurately models that portion of the transfer function of an EPR spectrometer that is associated with magnetic field modulation [7
]. This figure reveals that an increase in modulation amplitude leads not only to an increase in signal intensity, but also to an increase in spectral broadening. To measure lineshape parameters in a CW experiment, a compromise must be made to achieve sufficient signal intensity while maintaining spectral integrity. This lineshape-lineheight compromise is typically addressed by selecting a modulation amplitude that is approximately one-quarter of the narrowest linewidth. Spectral distortion is diminished under these conditions, but shows that more than 80% of the pure absorption signal is also sacrificed. The methods of this paper are designed to overcome this loss of an estimated factor of five in signal height.
(a) Simulated absorption spectrum of a nitroxide at 9.5 GHz where gx = 2.0085, gy = 2.0061, gz = 2.0023; Ax = 6.2 G, Ay = 4.3 G, Az = 36.9 G; and τR = 8 × 10−10 s.
The effects of field modulation presented in are well documented [8
], and methods of analysis have been developed [10
]. Alternative detection schemes, such as direct current (DC) detection, are not sensitive enough to observe weak signals due to phase and receiver noise, while alternative techniques, such as microwave frequency swept EPR or rapid scan EPR spectroscopy, deal with responses of the spin system resulting from adiabatic rapid passage [11
]. Fedin et al.
obtained absorption spectra using a multifrequency approach with a modulated longitudinal field [13
]. These various methods have limitations. In the judgment of the authors, there remains a need to develop a practical and versatile method to collect the undistorted pure absorption CW EPR spectrum.