*Paul T. Callaghan*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199556984
- eISBN:
- 9780191774928
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556984.003.0005
- Subject:
- Physics, Condensed Matter Physics / Materials, Nuclear and Plasma Physics

This chapter discusses magnetic field gradients and spin isochromats, and introduces their phase evolution under magnetic field gradients using the ideas of the magnetization helix and k-space, thus ...
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This chapter discusses magnetic field gradients and spin isochromats, and introduces their phase evolution under magnetic field gradients using the ideas of the magnetization helix and k-space, thus leading to a description of Magnetic Resonance Imaging and selective excitation. Translational motion encoding using various echoes is covered using both the spin jump model and the magnetization diffusion equation. The Stejskal-Tanner Pulsed Gradient Spin Echo method is discussed along with the role of background gradients in finite gradient pulse rise times. Using the neutron scattering analogy, q-space, and the propagator formalism, the narrow gradient pulse approximation is explained, leading to a discussion of the multiple propagator/matrix method for handling finite width gradient pulses. Frequency Domain-modulated gradient methods are presented along with the various approximations involved. Finally, homospoiling and phase cycles, and the use of RF field gradients for the measurement of translational motion, are discussed.Less

This chapter discusses magnetic field gradients and spin isochromats, and introduces their phase evolution under magnetic field gradients using the ideas of the magnetization helix and k-space, thus leading to a description of Magnetic Resonance Imaging and selective excitation. Translational motion encoding using various echoes is covered using both the spin jump model and the magnetization diffusion equation. The Stejskal-Tanner Pulsed Gradient Spin Echo method is discussed along with the role of background gradients in finite gradient pulse rise times. Using the neutron scattering analogy, q-space, and the propagator formalism, the narrow gradient pulse approximation is explained, leading to a discussion of the multiple propagator/matrix method for handling finite width gradient pulses. Frequency Domain-modulated gradient methods are presented along with the various approximations involved. Finally, homospoiling and phase cycles, and the use of RF field gradients for the measurement of translational motion, are discussed.

*Robert H. Swendsen*

- Published in print:
- 2019
- Published Online:
- February 2020
- ISBN:
- 9780198853237
- eISBN:
- 9780191887703
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198853237.003.0026
- Subject:
- Physics, Condensed Matter Physics / Materials, Theoretical, Computational, and Statistical Physics

In Chapter 26 we return to calculating the contributions to the specific heat of a crystal from the vibrations of the atoms. The vibrations of a model of a solid, for which the interactions are ...
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In Chapter 26 we return to calculating the contributions to the specific heat of a crystal from the vibrations of the atoms. The vibrations of a model of a solid, for which the interactions are quadratic in form, is investigated. Calculations are restricted to one dimension for simplicity in the derivations of the Fourier modes and the equations of motion. Both pinned and periodic boundary conditions are discussed. The representation of the Hamiltonian in terms of normal modes and the solution in terms of the equations of motion are derived. The Debye approximation is then introduced for three-dimensional systems.Less

In Chapter 26 we return to calculating the contributions to the specific heat of a crystal from the vibrations of the atoms. The vibrations of a model of a solid, for which the interactions are quadratic in form, is investigated. Calculations are restricted to one dimension for simplicity in the derivations of the Fourier modes and the equations of motion. Both pinned and periodic boundary conditions are discussed. The representation of the Hamiltonian in terms of normal modes and the solution in terms of the equations of motion are derived. The Debye approximation is then introduced for three-dimensional systems.