F20EPR
Measuring Relaxation Time Using Longitudinal Detection in Low-field Electron Paramagnetic Resonance Spectroscopy
Haoyue Sun, Chenyu Yu
Supervisor: Professor Michael Garwood, Professor Elias Puchner
Abstract
In this experiment, the longitudinal magnetization due to a transverse-plane radiofrequency field was studied. We used this phenomenon to improve traditional Electron Paramagnetic Resonance (EPR) spectroscopy by developing the longitudinal detection (LOD) system. LOD is capable of measuring electron spins with short relaxation through simultaneous transmit and receive. The Modified Bloch's Equation was used in attempts to describe the measured phenomena. Our main sample of study were iron oxide nanoparticles (IONP) and 2,2-diphenyl-1-picrylhydrazyl (DPPH). Such materials have short relaxation times, and therefore are difficult to characterize with conventional EPR spectroscopy. The Spectra without a constant polarization field was simulated and analyzed by the Full Width at Half Maximum (FWHM) method to interpret the relation between FWHM and relaxation time.
Introduction & Motivation
In a conventional EPR machine (Figure 1a), only one set of coils is used in magnetizing sample and detecting its relaxation process. In this setup, one has to turn one the coils to first polarize the sample (Transmit), and then switch the coils to detection mode to document the relaxation process of electron spins (Receive). Two problems emerge from this setup. It cannot measure short relaxation time (~nanosecond) as the coils need to be switched from transmit mode to receive mode, and thus simultaneous transmit and receive is not possible. In addition to that, noise from residual transmit signals may interfere with the receive process. In our LOD system (Figure 1b), one dedicated coils in the xy-plane are used to produce a circularly polarized radio-frequency field to magnetize electron spins. As there are two sets of coils for transmit and receive, simultaneous transmit and receive is achievable and short relaxation times are measurable. Furthermore, due to the orthogonality of these two sets of coils, noises from each other becomes less sensitive, resulting in a higher signal to noise ratio (SNR).
Figure 1. (a) An oversimplified schematics of a conventional EPR; (b) An rough depiction of our LOD system.
Theory
Our simulation is based on Modified Bloch's Equation [Whitfield1967]:
When there is an external field present, we can solve the modified Bloch's equation exactly for a steady-state Mz solution:
In the above equations, γ is the gyromagnetic ratio of the electron, is the strength of the radio-frequency field, H0 is the polarizing field magnitude. It is rather surprising that when H0 = 0, Mz ≠ 0. This is exactly what gives rise to the longitudinal magnetization phenomenon. To better understand this: we define the fictitious field by subtracting the contribution in Mz by H0, since we are interested to know what happens when
vanishes.
We then plug the steady-state solution into the definition of the fictitious field.
We remark that the fictitious field defined here is in part stemmed from a rotating frame of the system (the RF-field generated is circularly polarized, and thus itself is rotating around the z-axis, creating this fictitious field). This can be understood intuitively by considering a spin which relaxes back along the instantaneous RF field direction. While trying to relax, the spin is unable to follow the rotation of the RF field vector in xy-plane, hence at some later time, there will be an angle between the spin and RF field. Then the spin precesses around the RF field out of xy-plane, and thus contribute to a stationary magnetization perpendicular to xy-plane. From the above equation of the fictitious field, we see by setting H_0 to 0 [Tang2020].
If we modulate $H_f$, we have hope to produce a periodic magnetization pattern which can be detected through the receive coil. And the equation of fictitious field without polarization field shows that it is possible to modulate
by modulating the RF field applied. We choose to use a circularly polarized RF field modulated by a square wave. The circularly polarized RF field is the superposition of two sinusoidal waves:
And we modulate this RF field by turning it on and off at a modulation frequency
.
In order for our system to be practical, we want to determine at what RF field frequency we have the maximum longitudinal signal. We do so by taking the partial derivative of the fictitious field and equate it to 0 to solve for
which is the local extremum:
From the condition for the maximum signal we see that when an external field is not present and presumably , we have maximum signal amplitude when
which is later verified in the result section.
Simulation Setup
In this project, we simulated the longitudinal magnetization of DPPH with and without a constant polarizing field H0, when a transverse RF-field is applied. In our simulation, we solved the Modified Bloch's Equation numerically using the explicit fourth-order Runge-Kutta method [Zeigler1976]. The simulator takes the input of RF field vector along the direction of
direction, magnetic susceptibility and relaxation time constant of the sample to simulate, the time interval we wish to compute M(t) as well as the time step (or resolution) of the solution. The simulator computes the simulated M(t) during the specified time interval. In the result section, we demonstrate the simulation of DPPH at a temperature of 293K. The relaxation time constant of pure DPPH was measured to be
[Roest1959]. Here we note that can change when the concentration of DPPH is lower [Brown2014]. Due to the hardware setup in Garwood's lab, we use the following parameters for the RF field:
and the modulation frequency is .
Simulation Result
The response of longitudinal magnetization in DPPH sample to circularly RF field applied is given in figure 2.
Figure 2. The simulated magnetization pattern in DPPH when applying a circularly polarized RF field. The plot on the left is the RF field vector in x-direction and y-direction. The plot on the right is the magnetization response in z-direction in the sample.
By taking the time derivative of the magnetization in the z-direction, we obtain a simulation of the longitudinal signal detected in the receive coil, as the coil only detects the change of magnetization in time (figure 3.).
Figure 3. Since the receive coil only detects the first time derivative of the magnetization in the z-direction, so the time domain signal is the derivative of Figure 2 plot on the right.
Figure 4. We analyze the time domain signal at the moment when the circularly-polarized RF modulated with a square wave switches off. The Signal is a pure exponential decay. Hence we record the FWHM of the signal which can be shown to relate to the relaxation time of the sample.
The resolution of the signal in the time domain might cause the Fourier domain signal to shift away from 0 because the peak of the Lorentzian function might not have enough points. We show in figure 6 that the minimal resolution is to take 2.2*10^6 points in the time domain.
Figure 5. The plot of FWHM as a function of the inverse of relaxation time. The relationship is linear: .
Figure 6. The minimal resolution required for the R-squared of the linear fit of the relationship between FWHM and the inverse of relaxation time: to be 1. The minimal resolution is to take 2.2*10^6 points in the time domain.
Figure 6. The plot on the left is a signal perturbed with Gaussian random noise with variance 0.1T/s at all frequency. The plot on the right is the averaged then Fourier transformed signal with the original signal. The averaging technique can recover the signal almost exactly.
Discussion & Conclusion
Our simulations show that it is possible to directly measure short relaxation time using the phenomenon of longitudinal magnetization when there is no constant external polarization field present. The optimal RF field magnitude to use is
. And the minimum resolution required is points in time domain signal. If the minimal resolution is reached, the FWHM is related to relaxation time:
. The derivation can be found here: http://www.pci.tu-bs.de/aggericke/PC4e/Kap_III/Linienbreite.htm. One challenge to measure relaxation time as short as tens of nanoseconds in practice is that measuring shorter relaxation time means the FWHM of the measured signal will be broader and this means the receive bandwidth has to be broadened, hence the measurement might contain more noise. Our simulation also shows when the added noise is Gaussian white noise, we can recover the original signal around the peak exactly by averaging then fast Fourier transforming. In the future, we hope that the hardware for measuring relaxation time using LOD could be built and our simulation could be verified by experiment.
Acknowledgement
This project was in collaboration with Professor Michael Garwood from the Department of Radiology, and his PhD students--Xueyan Tang and Guhan Qian. We sincerely thank them for providing this opportunity, their tremendous support while we were conducting this project, as well as all the valuable discussions which helped us navigate through this project.
Reference
[1] F. Bloch and A. Siegert, Magnetic Resonance for Nonrotating Fields, Physics Review (1940)
[2] R.W. Brown et. al., Magnetic Resonance Imaging: Physical Principles and Sequence Design, 2nd Ed (2014)
[3] X. Tang, S. Suddarth, G. Qian, M. Garwood, Ultra-low frequency EPR using longitudinal detection and fictitious-field modulation, Journal of Magnetic Resonance, Vol 321, 2020, 106855
[4] G. Whitfield and A. G. Redfield, Paramagnetic Resonance Detection Along the Polarizing Field Direction, Physical Review (1957)
[5] R. Roest and N. J. Poulis, The spin-spin relaxation time of diphenyl picryl hydrazyl in weak fields, Antiviral Research 10.1016/0166-3542(59)90205-0 (1959).
[6] B. P. Zeigler, Theory of modelling and simulation. (1976).