Simple Derivation of the Bloch Equation

1966 ◽  
Vol 34 (12) ◽  
pp. 1164-1168 ◽  
Author(s):  
D. ter Haar
Keyword(s):  
Bloch Equation ◽  
Simple Derivation

scholarly journals Model forB1Imaging in MRI Using the Rotating RF Field

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Adnan Trakic ◽  
Jin Jin ◽  
Ewald Weber ◽  
Stuart Crozier
Keyword(s):  
Degrees Of Freedom ◽  
Acoustic Noise ◽  
Bloch Equation ◽  
Gradient Coil ◽  
Imaging Method ◽  
Nonuniform Field ◽  
Rf Coil ◽  
Image Encoding ◽  
Magnetic Resonance Imaging Mri ◽  
Space Requirements

Conventionally, magnetic resonance imaging (MRI) is performed by pulsing gradient coils, which invariably leads to strong acoustic noise, patient safety concerns due to induced currents, and costly power/space requirements. This modeling study investigates a new silent, gradient coil-free MR imaging method, in which a radiofrequency (RF) coil and its nonuniform field (B1+) are mechanically rotated about the patient. The advantage of the rotatingB1+field is that, for the first time, it provides a large number of degrees of freedom to aid a successfulB1+image encoding process. The mathematical modeling was performed using flip angle modulation as part of a finite-difference-based Bloch equation solver. Preliminary results suggest that representative MR images with intensity deviations of <5% from the original image can be obtained using rotating RF field approach. This method may open up new avenues towards anatomical and functional imaging in medicine.

scholarly journals Full analytical solution of the bloch equation when using a hyperbolic-secant driving function

2016 ◽  
Vol 77 (4) ◽  
pp. 1630-1638 ◽  
Author(s):  
Jinjin Zhang ◽  
Michael Garwood ◽  
Jang-Yeon Park
Keyword(s):  
Analytical Solution ◽  
Bloch Equation ◽  
Driving Function

GENERAL FORMULA FOR THE SECOND VIRIAL COEFFICIENT

1961 ◽  
Vol 39 (11) ◽  
pp. 1563-1572 ◽  
Author(s):  
J. Van Kranendonk
Keyword(s):  
Phase Shift ◽  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Resolvent Operators ◽  
Simple Derivation ◽  
Scattering Phase ◽  
Scattering Phase Shifts ◽  
Quantization Volume ◽  
Mechanical Expression ◽  
The Waves

A simple derivation is given of the quantum mechanical expression for the second virial coefficient in terms of the scattering phase shifts. The derivation does not require the introduction of a quantization volume and is based on the identity R(z)−R0(z) = R0(z)H1R(z), where R0(z) and R(z) are the resolvent operators corresponding to the unperturbed and total Hamiltonians H0 and H0 + H1 respectively. The derivation is valid in particular for a gas of excitons in a crystal for which the shape of the waves describing the relative motion of two excitons is not spherical, and, in general, varies with varying energy. The validity of the phase shift formula is demonstrated explicitly for this case by considering a quantization volume with a boundary the shape of which varies with the energy in such a way that for each energy the boundary is a surface of constant phase. The density of states prescribed by the phase shift formula is shown to result if the enclosed volume is required to be the same for all energies.

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