The application of spinors to solving the Bloch equations

Barratt, C.; Fackerell, Edward D.; Rosenfeld, Dov

doi:10.1016/0022-2364(89)90317-x

Cited by 9 publications

(5 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Inverting the Bloch equation has been well investigated for 1D, single-channel RF pulses. Some of the notable methods include the Shinnar-Le Roux method (32), analytic solution of some special forms of RF pulses (33), neural networks method (34), evolutionary method (35), simulated annealing (36), perturbation response based method (37), iterative correction for hardware nonlinearity (38), SPINCALC (39), inverse scattering transform method (40,41), stereographic projection method (42), and optimal control (43,44). Due to its flexibility for systematically imposing desirable constraints (e.g., RF power) and richness in efficient algorithms, the optimal control method has been applied to several applications to design improved 1D, single-channel RF pulses.…”

mentioning

confidence: 99%

Designing multichannel, multidimensional, arbitrary flip angle RF pulses using an optimal control approach

King

Zhu

et al. 2008

Magnetic Resonance in Med

125

View full text Add to dashboard Cite

The vast majority of parallel transmission RF pulse designs so far are based on small-tip-angle (STA) approximation of the Bloch equation. These methods can design only excitation pulses with small flip angles (e.g., 30 degrees ). The linear class large-tip-angle (LCLTA) method is able to design large-tip-angle parallel transmission pulses through concatenating a sequence of small-excitation pulses when certain k-space trajectories are used. However, both STA and LCLTA are linear approximations of the nonlinear Bloch equation. Therefore, distortions from the ideal magnetization profiles due to the higher order terms can appear in the final magnetization profiles. This issue is addressed in this work by formulating the multidimensional multichannel RF pulse design as an optimal control problem with multiple controls based directly on the Bloch equation. Necessary conditions for the optimal solution are derived and a first-order gradient optimization algorithm is used to iteratively solve the optimal control problem, where an existing pulse is used as an initial "guess." A systematic design procedure is also presented. Bloch simulation and phantom experimental results using various parallel transmission pulses (excitation, inversion, and refocusing) are shown to illustrate the effectiveness of the optimal control method in improving the spatial localization or homogeneity of the magnetization profiles.

show abstract

mentioning

confidence: 99%

Designing multichannel, multidimensional, arbitrary flip angle RF pulses using an optimal control approach

King

Zhu

et al. 2008

Magnetic Resonance in Med

125

View full text Add to dashboard Cite

show abstract

“…Then, as the offset of such pulse reaches the orientation-dependent resonance frequency f͑⍀͒ of a spin within the powder, this isochromat will become excited and begin its free precession. 16,37 As justified in further detail elsewhere, the overall phase exc collected by the spins under such circumstances can be accurately described by the sum of two terms. 35,36 One involves the phase accrued by the rf up to the time ͑f͒ when its offset has reached a value of f; the other describes the spin's ensuing precession under the assumption of an instantaneous excitation and a subsequent free evolution.…”

Section: A Quadrupolar Patterns By Fourier-transform Of Distortion-fmentioning

confidence: 90%

“…In other words, we explore whether the frequency-swept excitation/refocusing procedures just described can yield line shapes endowed with better SNRs than their conventional spin-echo counterparts, while operating under conditions that fulfill the weak rf regime required to obtain f-independent nutation angles. As detailed elsewhere, 26,27,35,37 the amplitudes required by a chirped rf to impart either a = /2 ͑nonadiabatic͒ or a = ͑adiabatic͒ rotation of the spins depend on the square root of the sweeping rate R; for instance, for a spin-1 / 2 nucleus, the nutation field required for a / 2 excitation is 1 Ϸ 0.25 ͱ R exc , while an in-plane -type refocusing requires 1 ജ 0.8 ͱ R ref .…”

Section: A Quadrupolar Patterns By Fourier-transform Of Distortion-fmentioning

confidence: 99%

Quadrupolar nuclear magnetic resonance spectroscopy in solids using frequency-swept echoing pulses

Bhattacharyya

Frydman

2007

The Journal of Chemical Physics

122

128

View full text Add to dashboard Cite

The acquisition of ideal powder line shapes remains a recurring challenge in solid-state wideline nuclear magnetic resonance (NMR). Certain species, particularly quadrupolar spins in sites associated with large electric field gradients, are difficult to excite uniformly and with good efficiencies. This paper discusses some of the opportunities that arise upon departing from standard spin-echo excitation approaches and switching to echo sequences that use low-power, frequency-swept radio frequency (rf) pulses instead. The reduced powers demanded by such swept rf fields allow one to excite spins in different crystallites efficiently and with orientation-independent pulse angles, while the large bandwidths of interest that are needed by the measurement can be covered, thanks to the use of broadband frequency sweeps. The fact that the spins' evolution and ensuing dephasing starts at the beginning of such rf manipulation calls for the use of spin-echo sequences; a number of alternatives capable of providing the desired line shapes both in the frequency and in the time domains are introduced and experimentally demonstrated. Sensitivity- and lineshape-wise these experiments are competitive vis-a-vis current implementations of wideline quadrupolar NMR based on hard rf pulses; additional opportunities that may derive from these ideas are also briefly discussed.

show abstract

“…Thus we find the fundamental system in terms of m (i) (i = 1, 3, 4) . For later convenience, from this point, we replace m (3) and m (4) by m (2) and m (3) , respectively. We thus define the fundamental matrix K by…”

Section: A Fundamental System Of the Homogeneous Bloch Equationmentioning

confidence: 99%

A Solution of the Inhomogeneous Bloch Equation for a Class of Time-Varying Magnetic Fields

Kobayashi

2004

Progress of Theoretical Physics

View full text Add to dashboard Cite

A solution of the inhomogeneous Bloch equation is given for a class of time-varying magnetic fields. A method of determining the fundamental system of the homogeneous Bloch equation employing the characteristics of the Riccati equation is formulated. It turns out that the fundamental matrix is an orthogonal matrix. A brief discussion of the characteristics of the magnetic field class and an illustrative example are given. §1. IntroductionSince the Bloch equation 1) was proposed in 1946, many people have attempted to solve it analytically. However, as yet, few exact solutions are known. One method of solving the homogeneous Bloch equation is the spinor approach, in which the vector Bloch equation is spinorized. 2), 3) There exists a 2 × 2 unitary evolution matrix under which the magnitude of the spinor is conserved. Keeping this point in mind, we developed a vector approach by using the proper orthogonal transformation (rotation). 4) With this approach, we were able to obtain a great deal of information regarding exact solutions of the homogeneous Bloch equation.Recently, we found that the homogeneous vector Bloch equation can be reduced to the Riccati equation by introducing two complex variables. 5) By solving the Riccati equation, we succeeded in finding a set of three independent solutions of the homogeneous Bloch equation, which is called the fundamental system of the homogeneous system of three first-order linear differential equations, i.e., the homogeneous Bloch equation. 6) The spinorized Bloch equation can be cast in the form of a Schrödinger equation in which the free and interaction Hamiltonians correspond to the terms involving the third component of the magnetic fields and the other two components, respectively. 7) This fact strongly suggests that the homogeneous Bloch equation cannot be solved in general. In other words, it seems that the homogeneous Bloch equation can be solved only in certain restricted cases. Considering this point, the method developed by Mitrinovitch 8) and Guigue 9) is the most appropriate one for solving the homogeneous Bloch equation.The purpose of this article is to obtain a physically acceptable solution of the inhomogenenous Bloch equation. This equation is uniquely solved by finding a fundamental system of the homogeneous Bloch equation, as is well known. It turns out that the fundamental matrix is an orthogonal matrix.Using the method mentioned above, we find a special solution of the homogeneous Bloch equation. Then, further special solutions that form a fundamental

show abstract

The application of spinors to solving the Bloch equations

Cited by 9 publications

References 1 publication

Designing multichannel, multidimensional, arbitrary flip angle RF pulses using an optimal control approach

Designing multichannel, multidimensional, arbitrary flip angle RF pulses using an optimal control approach

Quadrupolar nuclear magnetic resonance spectroscopy in solids using frequency-swept echoing pulses

A Solution of the Inhomogeneous Bloch Equation for a Class of Time-Varying Magnetic Fields

Contact Info

Product

Resources

About