Helmut Hofmann
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198504016
- eISBN:
- 9780191708480
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504016.003.0026
- Subject:
- Physics, Nuclear and Plasma Physics
This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, ...
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This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, representations of the delta-function are given and applied to Fourier and Laplace transformations. For quantal operators, the Mori product is introduced and an important formula for the derivative of exponentials is shown. Elementary properties of spin and isospin are discussed; for fermions, the formalism of second quantization is produced.Less
This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, representations of the delta-function are given and applied to Fourier and Laplace transformations. For quantal operators, the Mori product is introduced and an important formula for the derivative of exponentials is shown. Elementary properties of spin and isospin are discussed; for fermions, the formalism of second quantization is produced.
Kyösti Kontturi, Lasse Murtomäki, and José A. Manzanares
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199533817
- eISBN:
- 9780191714825
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199533817.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter is concerned with transport in the vicinity of electrodes, and hence on the coupling between Faradaic electrode processes and mass transport. It covers transport in stationary and ...
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This chapter is concerned with transport in the vicinity of electrodes, and hence on the coupling between Faradaic electrode processes and mass transport. It covers transport in stationary and transient condition, planar and spherical geometries, presence and absence of supporting electrolytes, as well as convective transport in hydrodynamic electrodes. Some common electrochemical techniques are also discussed, and the solutions of the corresponding transient transport problems are worked out in detail.Less
This chapter is concerned with transport in the vicinity of electrodes, and hence on the coupling between Faradaic electrode processes and mass transport. It covers transport in stationary and transient condition, planar and spherical geometries, presence and absence of supporting electrolytes, as well as convective transport in hydrodynamic electrodes. Some common electrochemical techniques are also discussed, and the solutions of the corresponding transient transport problems are worked out in detail.
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/9780199556984.003.0009
- Subject:
- Physics, Condensed Matter Physics / Materials, Nuclear and Plasma Physics
This chapter shows how both Fourier transformation and inverse Laplace transformation can be used in conjunction with pulsed gradient spin echo NMR, as a basis for a multidimensional ...
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This chapter shows how both Fourier transformation and inverse Laplace transformation can be used in conjunction with pulsed gradient spin echo NMR, as a basis for a multidimensional characterisation, whether as separation, correlation, or exchange experiments in which diffusion and relaxation effects provide additional dimensions. In a porous medium, exchange experiments reveal changes in local relaxation rates and diffusion coefficients caused by molecular migration, and when measured as a function of mixing time can reveal exchange times. At zero mixing time, separate properties such as diffusion, local field, local field gradient, or relaxation time may be correlated. The chapter explains the details of inverse Laplace transformation, and takes the reader through the various multiplexed Fourier and Laplace dimensions in which molecular translation plays a role. Finally, measurement of the diffusion tensor is discussed as an example of multidimensional signal acquisition and processing.Less
This chapter shows how both Fourier transformation and inverse Laplace transformation can be used in conjunction with pulsed gradient spin echo NMR, as a basis for a multidimensional characterisation, whether as separation, correlation, or exchange experiments in which diffusion and relaxation effects provide additional dimensions. In a porous medium, exchange experiments reveal changes in local relaxation rates and diffusion coefficients caused by molecular migration, and when measured as a function of mixing time can reveal exchange times. At zero mixing time, separate properties such as diffusion, local field, local field gradient, or relaxation time may be correlated. The chapter explains the details of inverse Laplace transformation, and takes the reader through the various multiplexed Fourier and Laplace dimensions in which molecular translation plays a role. Finally, measurement of the diffusion tensor is discussed as an example of multidimensional signal acquisition and processing.
Bernhard Blümich
- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198526766
- eISBN:
- 9780191709524
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526766.003.0004
- Subject:
- Physics, Condensed Matter Physics / Materials
Transformation, convolution, and correlation are used over and over again in nuclear magnetic resonance (NMR) spectroscopy and imaging in different contexts and sometimes with different meanings. The ...
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Transformation, convolution, and correlation are used over and over again in nuclear magnetic resonance (NMR) spectroscopy and imaging in different contexts and sometimes with different meanings. The transformation best known in NMR is the Fourier transformation in one or more dimensions. It is used to generate one- and multi-dimensional spectra from experimental data as well as ID, 2D, and 3D images. Furthermore, different types of multi-dimensional spectra are explicitly called correlation spectra. These are related to nonlinear correlation functions of excitation and response. This chapter discusses convolution in linear and nonlinear systems, along with the convolution theorem, linear system analysis, nonlinear cross-correlation, correlation theorem, Laplace transformation, Hankel transformation, Abel transformation, z transformation, Hadamard transformation, and wavelet transformation.Less
Transformation, convolution, and correlation are used over and over again in nuclear magnetic resonance (NMR) spectroscopy and imaging in different contexts and sometimes with different meanings. The transformation best known in NMR is the Fourier transformation in one or more dimensions. It is used to generate one- and multi-dimensional spectra from experimental data as well as ID, 2D, and 3D images. Furthermore, different types of multi-dimensional spectra are explicitly called correlation spectra. These are related to nonlinear correlation functions of excitation and response. This chapter discusses convolution in linear and nonlinear systems, along with the convolution theorem, linear system analysis, nonlinear cross-correlation, correlation theorem, Laplace transformation, Hankel transformation, Abel transformation, z transformation, Hadamard transformation, and wavelet transformation.
