Bloch vector model

To verify the theory of PIP, a computer program using C language was developed. It can be used to directly calculate the excitation profiles by PIPs or any other RF pulses. The calculation is based on the Bloch vector model for a non-interacting spin-1/2 system, where the spin-lattice relaxation during the pulse is neglected. The basic idea of the program is discussed as follows. 

The Rydberg atom experiments described above are well adapted to the study of the atomic observables via the very sensitive field ionization method. The observation of the field itself and its fluctuations would also be very interesting. (In the Bloch vector model, the field variables are associated to the pendulum velocity whereas the atomic ones are related to its position). It has recently been shown either by full quantum mechanical calculations or by the Bloch vector semi-classical approach that if the system is initially triggered by a small external field impinging on the cavity, the fluctuations on one phase of the field become at some time smaller than in the vacuum field. This is a case of radiation "squeezing" which would be very interesting to study on Rydberg atom maser systems. 

This section summarizes primarily the classical description of NMR based on the vector model of the Bloch equations. Important concepts like the rotating frame, the effect of rf pulses, and the free precession of transverse magnetization are introduced. More detailed accounts, still on an elementary level, are provided in textbooks [Deri, Farl, Fukl]. 

Most imaging techniques can be understood within the vector model of the Bloch equations (cf. Section 2.2.1). For this reason, the magnetization response calculated from the Bloch equations is investigated for arbitrary rf input and arbitrarily time-dependent magnetic-field gradients. In particular, the response which depends linearly on the rf excitation is of interest not only for imaging the spin density Mo(r), but also for... 

We consider here a particular model for the Detector which fulfills several purposes. A 2-state Detector, or Detecton, is the simplest possible quantal device that can probe which-way information of the Quanton. The DETECTON can then be viewed as a QUANTON itself, likewise describable by a predictability Vd and a fringe visiblity Vdq-Its initial state can also be described by a Bloch vector which can, if necessary, be... 

The spin dynamics of the deuteron (spin /= 1) are more complex than those of the spin 1/2 nuclei, and the simple vector model used in other chapters, derived from the Bloch equations, provides no particular insight into deuteron spin dynamics. However, some of the geometric simplicity of the Bloch equations is present in a product-operator formalism, used to describe spin 1 NMR [117]. This formalism can provide a visual understanding of the deuteron pulse sequences in terms of simple precession and pulse rotations, albeit among a greater number of coordinate axes. The formalism can be used to understand the production of quadrupole order and the T q relaxation time (Figure 8.2(b)) and the two-dimensional deuteron exchange experiment (section 8.5). 

An elastic collision may be described, in the Bloch vector picture, by a "jump" of the polarization vector P on its precession cone, such that its projection along the w axis remains unchanged. One may describe such a collision by transformation of rotation by an angle 9. The evolution of the angle of rotation 6 depends on the particular model assumed for the collision. In the impact approximation where the duration of the... 

As a potentially appealing application of these ideas, we proceed to consider the possibilities for lasing action in a field mode on-resonance with channeling radiation, propagating alongside a beam of channeled electrons. Due to the anharmonicity of the channeling potential for electrons, the o)eg °f different transitions are well-separated and the two-level model for emission applies. Define a cooperative Bloch-vector density for this system... 

Here the dielectric permittivity is spatially periodic, e(x + ti) = e(x), which means that the Kronig-Penney model is applicable. If we introduce the Bloch vector, i.e., the crystal momentum k, the periodicity of the electric field will be described by E x + d) = e E x)—Bloch or Floquet condition. The solution of the wave equation for an infinite ID lattice with periodically changing dielectric permittivity should have the form of a sum of a direct and a reflected wave... 

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

For a given sequence, Bloch equations give the relationship between the explanatory variables, x, and the true response, i]. The / -dimensional vector, 0, corresponds to the unknown parameters that have to be estimated x stands for the m-dimensional vector of experimental factors, i.e., the sequence parameters, that have an effect on the response. These factors may be scalar (m — 1), as previously described in the TVmapping protocol, or vector (m > 1) e.g., the direction of diffusion gradients in a diffusion tensor experiment.2 The model >](x 0) is generally non-linear and depends on the considered sequence. Non-linearity is due to the dependence of at least one first derivative 5 (x 0)/50, on the value of at least one parameter, 6t. The model integrates intrinsic parameters of the tissue (e.g., relaxation times, apparent diffusion coefficient), and also experimental nuclear magnetic resonance (NMR) factors which are not sufficiently controlled and so are unknown. 

Further development of Sommerfeld s theory of metals would extend well outside the intended scope of this textbook. The interested reader may refer to any of several books for this (e.g. Seitz, 1940). Rather, this book will discuss the band approximation based upon the Bloch scheme. In the Bloch scheme, Sommerfeld s model corresponds to an empty lattice, in which the electronic Hamiltonian contains only the electron kinetic-energy term. The lattice potential is assumed constant, and taken to be zero, without any loss of generality. The solutions of the time-independent Schrodinger equation in this case can be written as simple plane waves, = exp[/A r]. As the wave function does not change if one adds an arbitrary reciprocal-lattice vector, G, to the wave vector, k, BZ symmetry may be superimposed on the plane waves to reduce the number of wave vectors that must be considered ... 

Kordower JH, Emborg ME, Bloch J, Ma SY, Chu Y, Leventlial L, McBride J, Chen EY, Palfi S, Roitberg BZ, Brown WD, Holden JE, Pyzalski R, Taylor MD, Caivey P, Ling Z, Trono D, Hand aye P, Deglon N, Aebischer P (2000) Neui odegeneradon prevented by lendviral vector delivery of GDNF in primate models of Pai kin-son s disease. Science 290 767—773. 

Kordower JH, Emborg ME, Bloch J, Ma S Y, Chu Y, et al. 2000. Neurodegeneration prevented by lentiviral vector delivery of GDNF in primate models of Parkinson s disease. Science 290 767-73... 

Example 1 Most discussions involving the Bloch model introduce the concept of the rotating frame. The concept of a rotating coordinate system is a familiar one because in real life positions and motion are referred to the earth, a coordinate system that is rotating. Similarly rather than refer the motion of the magnetization vectors to the fixed laboratory coordinate system, it is simpler to refer their motion to a rotating frame of reference which rotates at the NMR transmitter frequency of the nucleus under study. 

Pseudo ID crystals are often described by a tight binding Hubbard type model with matrix elements t j and tj. (t >>tj, ), an intramolecular Coulomb interaction, U, and an electron-phonon interaction parameter, X. The quasiparticle states of the electrons can be described by the usual Bloch model with wave vector, k, provided that U is sufficiently small compared with the band width, At. However, because of phase space considerations, particle-particle scattering is much more important in ID than in 3D systems. 

In general, we will have to assume that the 5 and 5° spaces are not orthogonal. This means that there does not exist a vector in the 5-space which is orthogonal to all of the vectors of the 5°-space (22). In addition, the states )° constitute a nonorthonormal basis set for the model space, 5°. From a physical point of view, it is important to have a one-to-one correspondence between the exact eigenvectors, 1 ), and the vectors )°. However, another basis set, denoted w)°, n = 1,, biorthogonal to the previous one w)°, n = 1, N has to be defined and used in Bloch s formulation. These vectors satisfy the following equations ... 

As described earlier, electronic states of atoms are dispersed in energy bands when condensed into a crystalline solid phase. The descriptive model chosen allows a qualitative understanding of the origins of bands, but neglects many details such as the effects of atomic periodicity and crystal symmetry. Quantitative treatment of an allowed electronic state in a solid ( /) is based on Bloch s theorem which states that /(r) = e M(r) where r is a location in the unit cell, k is the wave vector, and (r) renresents a periodic electrostatic potential (see Ashcroft and Mermin 1976). The term... 

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