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E-grāmata: Essential Mathematics for NMR and MRI Spectroscopists

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(Retired University of Saskatchewan, Canada)
  • Formāts: 867 pages
  • Izdošanas datums: 28-Aug-2020
  • Izdevniecība: Royal Society of Chemistry
  • Valoda: eng
  • ISBN-13: 9781839162961
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Essential Mathematics for NMR and MRI SpectroscopistsPalielināt
  • Formāts: 867 pages
  • Izdošanas datums: 28-Aug-2020
  • Izdevniecība: Royal Society of Chemistry
  • Valoda: eng
  • ISBN-13: 9781839162961
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Beginning with a review of the important areas of mathematics, this book then covers many of the underlying theoretical and practical aspects of NMR spectroscopy from a maths point of view. Competence in algebra and introductory calculus is needed but all other maths concepts are covered. It will bridge a gap between high level and introductory titles used in NMR spectroscopy. Uniquely, it takes a very careful and pedagogical approach to the mathematics behind NMR. It leaves out very few steps, which distinguishes it from other books in the field.

The author is an NMR laboratory manager and is sympathetic to the frustrations of trying to understand where some of the fundamental equations come from hence his desire to either explicitly derive all equations for the reader or direct them to derivations. This is an essential text aimed at graduate students who are beginning their careers in NMR spectroscopy and laboratory managers if they need an understanding of the theoretical foundations of the technique.



Beginning with a review of the important areas of mathematics, this book then covers many of the underlying theoretical and practical aspects of NMR spectroscopy from a maths point of view. This is an essential text aimed at graduate students and laboratory managers.

Recenzijas

Excellent for many aspects of NMR spectroscopy, including the treatment of product operator formalism and density matrix. * University of Minnesota *

1 Complex Numbers 1(11)
1.1 The History of Numbers
1(1)
1.2 Why Complex Numbers?
2(3)
1.3 Properties of Complex Numbers
5(6)
Further Reading
11(1)
2 Matrices 12(39)
2.1 Matrix Algebra
12(7)
2.2 Determinants
19(8)
2.3 Solving Matrix Equations
27(13)
2.4 Transforms
40(7)
2.5 Exponentials of Matrices
47(3)
Further Reading
50(1)
3 Vectors 51(68)
3.1 A Preliminary Definition of Vectors
51(4)
3.2 Euclidean Vector Algebra and Inner Product Spaces
55(5)
3.3 Vector Multiplication
60(18)
3.4 Matrix Notation
78(8)
3.5 Vector Transformations
86(9)
3.6 Dirac Notation and Operator Matrices
95(16)
3.7 Hilbert Space
111(1)
3.8 Merging Vector Spaces
112(6)
Further Reading
118(1)
4 Vector Calculus 119(58)
4.1 Vector Differentiation
119(9)
4.2 Vector Integration
128(21)
4.3 Curvilinear Coordinates and Grad, Div and Curl
149(8)
4.4 Green's Theorem, Gauss's Theorem and Stokes' Theorem
157(16)
4.5 Vector Rotation in a Rotating Frame of Reference
173(3)
Further Reading
176(1)
5 Introductory Tensor Analysis 177(20)
5.1 Dyadic Algebra
177(8)
5.2 The Gradient of a Vector
185(1)
5.3 Transformations
186(2)
5.4 A Tensor Definition
188(2)
5.5 Isotropic Tensors
190(4)
5.6 Symmetric, Antisymmetric and Irreducible Tensors
194(2)
Further Reading
196(1)
6 Probability and Statistics 197(23)
6.1 Simple Probability and Statistics
197(3)
6.2 Permutations and Combinations
200(4)
6.3 Stirling's Approximation
204(1)
6.4 Probability Distributions
205(2)
6.5 The Boltzmann Distribution
207(10)
6.6 Random Processes and the Autocorrelation Function
217(2)
Further Reading
219(1)
7 Waves 220(15)
7.1 Characteristics of Waves
220(3)
7.2 The Classical Wave Equation
223(8)
7.3 Superposition of Waves
231(1)
7.4 Standing Waves
232(2)
Further Reading
234(1)
8 The Fourier Transform 235(70)
8.1 Fourier Series
235(8)
8.2 Properties of the Time-Domain Functions
243(5)
8.3 The Fourier Integral
248(2)
8.4 Properties of Fourier Transforms
250(3)
8.5 Some Important Fourier Pairs
253(15)
8.6 Convolution
268(6)
8.7 Impulse Sampling and the Nyquist Frequency
274(5)
8.8 The Discrete Fourier Transform
279(8)
8.9 Real and Complex DFTs
287(7)
8.10 The Fast Fourier Transform
294(10)
Further Reading
304(1)
9 Introductory Classical Mechanics 305(22)
9.1 Newton's Laws
305(5)
9.2 Energy
310(2)
9.3 Generalised Coordinates
312(1)
9.4 The Lagrangian
313(5)
9.5 The Hamiltonian
318(1)
9.6 Classical Angular Momentum
319(7)
Further Reading
326(1)
10 Special Relativity 327(22)
10.1 The Lorentz Transformation Equations
327(9)
10.2 Kinematic Consequences of the Lorentz Transformations
336(3)
10.3 Space-time
339(5)
10.4 Minkowski Diagrams
344(4)
Further Reading
348(1)
11 Electric and Magnetic Fields 349(63)
11.1 Coulomb's Law and the Electric Field
349(3)
11.2 Field Flux and Gauss's Law
352(7)
11.3 The Magnetic Field
359(18)
11.4 Maxwell's Laws
377(5)
11.5 Special Relativity and Magnetism
382(11)
11.6 Electric Dipoles
393(6)
11.7 Magnetic Dipoles
399(12)
Further Reading
411(1)
12 The Magnetic Moment and the Bloch Equations 412(23)
12.1 The Bloch Equations
412(15)
12.2 The Vector Model of NMR Spectroscopy
427(6)
Further Reading
433(2)
13 Introductory Quantum Mechanics 435(25)
13.1 Historical
435(9)
13.2 The Basics of Quantum Mechanics
444(2)
13.3 The Postulates of Quantum Mechanics
446(5)
13.4 Vector Notation
451(1)
13.5 The Uncertainty Principle
452(7)
Further Reading
459(1)
14 The Hamiltonians of NMR 460(9)
14.1 The NMR Hamiltonian
460(1)
14.2 The Zeeman Field Hamiltonian
461(2)
14.3 The Chemical Shift Hamiltonian
463(2)
14.4 The Dipolar Coupling Hamiltonian
465(1)
14.5 The Indirect (Scalar) Coupling Hamiltonian
465(1)
14.6 The RF-Field Hamiltonian
466(2)
Further Reading
468(1)
15 The Quantum Mechanics of Angular Momentum 469(30)
15.1 A Brief History
469(1)
15.2 General Angular Momentum
470(11)
15.3 Angular Momentum Operators in Spherical Coordinates
481(2)
15.4 Angular Momentum Operator Matrices
483(5)
15.5 Expectation Values
488(2)
15.6 Time Evolution
490(8)
Further Reading
498(1)
16 NMR Rotation Operators 499(11)
16.1 Rotation in Euclidean Space
499(3)
16.2 The Euler Rotation Matrix
502(4)
16.3 Angular Momentum Rotation Operators
506(3)
Further Reading
509(1)
17 Density Operators and Matrices 510(51)
17.1 The Density Matrix
510(4)
17.2 Pure and Mixed States
514(4)
17.3 The Time Evolution of the Density Matrix
518(9)
17.4 Multiple-Spin Density Matrices
527(16)
17.5 Spin-One
543(6)
17.6 Density Matrices and Perturbation Calculations
549(4)
17.7 Strong Coupling
553(7)
Further Reading
560(1)
18 The Product Operator Formalism 561(37)
18.1 The Density Operator Method Revisited
561(2)
18.2 The Product Operator Method
563(5)
18.3 Multiple Quantum Coherence
568(2)
18.4 Product Operator Rotations
570(3)
18.5 Pulse Sequence Analysis
573(7)
18.6 Three Spins and Higher
580(8)
18.7 The Product Operator Tree
588(4)
18.8 Single Transition Product Operators
592(5)
Further Reading
597(1)
19 Coherence Transfer 598(38)
19.1 Phase Cycling
598(5)
19.2 Coherence Order
603(9)
19.3 Coherence Transfer Pathways
612(5)
19.4 Phase Cycling Revisited
617(9)
19.5 Pulsed Field Gradients
626(9)
Further Reading
635(1)
20 Relaxation and the NOE 636(34)
20.1 Spin Temperature
636(2)
20.2 Longitudinal T1 Relaxation
638(2)
20.3 Transverse T2 Relaxation
640(3)
20.4 How Does Relaxation Work?
643(18)
20.5 The Nuclear Overhauser Enhancement
661(7)
Further Reading
668(2)
21 Data Processing 670(31)
21.1 Fourier Transform Spectroscopy
670(9)
21.2 The 2D Fourier Transform
679(3)
21.3 Window Functions
682(3)
21.4 Zero Filling
685(4)
21.5 Digital Filtering
689(2)
21.6 Oversampling
691(1)
21.7 Phase Correction
692(3)
21.8 Linear Prediction
695(3)
21.9 Baseline Correction
698(2)
Further Reading
700(1)
22 Magnetic Resonance Imaging 701(28)
22.1 Selective Pulses
701(1)
22.2 Gradients
702(9)
22.3 Putting It All Together
711(4)
22.4 The Discrete Fourier Transform
715(3)
22.5 T1, T2 and Spin Density Weighting
718(5)
22.6 Pulse Sequences
723(5)
Further Reading
728(1)
23 Electronics 729(57)
23.1 The Modern NMR Spectrometer
729(2)
23.2 DC Electronics
731(16)
23.3 AC Electronics
747(9)
23.4 AC Circuits
756(16)
23.5 Antenna Theory
772(10)
23.6 Analogue-to-Digital Conversion
782(3)
Further Reading
785(1)
Appendix A Identities, Proofs and Miscellany 786(53)
A.1 Euler's Formula
786(2)
A.2 The Binomial Expansion
788(3)
A.3 Trigonometric Identities
791(6)
A.4 Direction Cosines
797(2)
A.5 The Area of a Parallelogram
799(1)
A.6 The Area of a Trapezoid
800(1)
A.7 The Number 'e'
801(2)
A.8 Alternate Coordinate Systems
803(8)
A.9 The Fundamental Theorem of Calculus
811(4)
A.10 Some Commonly Used Derivatives and Their Properties
815(1)
A.11 Partial Derivatives
816(1)
A.12 Parametric Equations
817(2)
A.13 Some Commonly Used Integrals and Their Properties
819(1)
A.14 Integration by Substitution
819(1)
A.15 Integration by Parts
820(1)
A.16 The Trapezoid Rule for Estimating Definite Integrals
821(2)
A.17 Solving Differential Equations
823(4)
A.18 Commutator Algebra
827(2)
A.19 L'Hopital's Rule
829(2)
A.20 Even and Odd Functions
831(3)
A.21 Group Theory
834(5)
Appendix B A Bruker to Varian Translation Table 839(1)
Subject Index 840
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