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Lectures on spin dynamics: the theoretical minimum
No full textThis book consists of a series of lectures introducing what the author believes to be the theoretical minimum for the understanding of nuclear spin dynamics, the branch of physics underpinning magnetic resonance techniques such as NMR and MRI. Spin dynamics ultimately concern the study of the state and time evolution of systems made by a large collection of particles possessing spin, one of the most subtle and indeed fascinating concepts in relativistic quantum mechanics and whose understanding requires a mix of quantum mechanics, Hamiltonian dynamics and advanced mathematics.Written in the concise and direct style appropriate for university lectures, this book is addressed to both undergraduate and postgraduate students who are approaching magnetic resonance studies and want to reach the theoretical minimum required to understand the wider topic and its main applications. The book is suitable for researchers who work in the field of magnetic resonance and want to know more about its theoretical fundamentals. Finally, built as a complete set of 12 lectures and 4 workshops, the book can act as a solid reference to lecturers of magnetic resonance university modules.Trained in a world-leading research laboratory internationally recognised for its contributions to the NMR field, the author has a track record in theoretical and methodological developments of nuclear spin dynamics. In this book, he tries to merge his passion for the understanding of physics through its mathematical rendering with his personal quest for the elegance of a clean, clear and satisfactory explanation
Singlet state relaxation via intermolecular dipolar coupling
No full textThe intermolecular contribution to the relaxation of singlet states has been derived on the basis of a translational-rotational diffusion model that describes molecules as impenetrable spheres which translate and rotate in an isotropic low-viscosity medium. The equations for the relaxation rate constants obtained are discussed and the dependence on physical parameters is exploited. Theoretical predictions are compared with experiments when the intermolecular relaxation is due to both protons and deuterons present in the sample. An agreement between experiments and theory of 4 was obtained when the physical parameters are estimated from first-principles calculation.</p
Singlet state relaxation via scalar coupling of the second kind
No full textThe contribution of scalar coupling relaxation of the second kind on the relaxation behaviour of nuclear spin singlet states has been derived. The analytical equation found for the relaxation rate constant of singlet state has been compared to the equation for the relaxation of longitudinal magnetization in order to find the conditions for which the singlet state remains long-lived even in the presence of this scalar relaxation mechanism. These results are relevant when the singlet state is formed in molecules with more than two interacting spins.</p
Chapter 3. Relaxation theory of long-lived spin order
No full textThis chapter discusses several relaxation mechanisms that contribute to the relaxation decay rate of singlet and longitudinal order. In a systematic approach I use the Redfield theory to obtain analytical equations for these contributions. The mechanisms that occur in isolated two-spin-1/2 pairs are discussed first and apply to cases where the singlet pair is the only spin system in the molecule. When other spins are present, other mechanisms may arise and some of these are discussed below. Finally, coherent leakage mechanisms that contribute to singlet but not to longitudinal order relaxation decay rates are introduced and discussed
Lecture 1: operator and superoperator algebra
No full textIn this lecture, I discuss some basic concepts of quantum mechanics, the ones that are most relevant to this series of lectures. I revise the concepts of states, operators and superoperators and the most important elements of their algebra, including traces, commutators, eigenvalues and eigensystems, observables and expectation values, basis sets and related concepts
Lecture 11: the dynamics of an ensemble of coupled spin-1/2 pairs
No full textIn this lecture, I discuss the dynamics of a large collection of identical spin systems made up of two coupled spin-1/2 nuclei. The discussion is restricted to a homonuclear pair (two spins of the same kind, hence, the same magnetogyric ratio) and the sample is assumed to be in an isotropic liquid state
Workshop 1: spin states and operators
No full textIn this workshop, the reader can practice with the use of basis sets, matrix diagonalization, calculating the eigenvalues of a Hamiltonian, deriving the matrix representation of spin operators and with the normalization of spin states and operators
Lecture 2: spin
No full textIn this lecture, I discuss the nature of spin in elementary particles as a consequence of relativistic corrections to quantum mechanics. I summarise the arguments introduced by Paul Dirac to derive the electron spin Hamiltonian and its eigenstates and eigenvalues. I then discuss the possible values of the spin quantum number of elementary particles and atomic nuclei
Lecture 10: the dynamics of a single spin-1/2 ensemble
No full textIn this lecture, I discuss the dynamics of a large collection of identical systems, each made up by a single spin-1/2 nucleus, in an isotropic liquid state. I show how to use knowledge gathered in previous lectures to calculate the state of such an ensemble either at thermal equilibrium, or at a later point in time after the system has been exposed to interactions with static and radiofrequency fields
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