Spin parameters, definitions

The word particle (e.g. nuclide or electron) is used routinely, and has ascribed to it measured properties such as total energy E, as well as spin parameters / and Mj. However its actual definition is fraught with danger (particle/wave conundrum, and its sub-structure mystery), and the concept of its exact size is nebulous. Its electrical charge too does not bear too close a scrutiny, because of its potential sub-structure composed of other smaller particles. It seems clear that all the particles in our Universe interact, at some range, to some extent. 

These definitions of spin parameters allow the computation of the various different coupled states present for any set of four individual spin states under the Hamiltonian (3), and the energies of these are given by Eq. (10). 

In the first section of this chapter spin systems and pulse sequences are discussed. Starting with the definition of a spin system and a description of the various spin parameters used by NMR-SIM. Using suitable examples the correct definition of spin system parameters, how to reduce the number of spin system parameters and the application of variable spin system parameters is illustrated. Following on, pulse sequences are discussed in some detail and the difference between a pulse sequence and a pulse program is emphasized. The pulse program language of NMR-SIM should be familiar to users of Bruker NMR spectrometers as it uses the same syntax. However, because this is not covered in the NMR-SIM manual, pulse sequence elements such as pulses, phases and delays are explained in detail and illustrated using a variety of examples. 

The labels used in the definition of the NMR-SIM spin system parameters depend to some extent upon personal preference. As with any programming language it is important to include comments comments not only makes the spin system file more readable, they also make it easier to keep track of any modifications to the file and help other users to understand the spin system definitions. Comment lines start with a semicolon ( ) and both the semicolon and the following text are ignored by the NMR-SIM processor. 

The first term is characterized by a scalar, 7, and it is the dominant term. Be aware of a convention disagreement in the definition of this term instead of -27, some authors write -7, or 7, or 27, and a mistake in sign definition will turn the whole scheme of spin levels upside down (see below). The second and third term are induced by anisotropic spin-orbit coupling, and their weight is predicted to be of order Ag/ge and (Ag/ge)2, respectively (Moriya 1960), when Ag is the (anisotropic) deviation from the free electron -value. The D in the second term has nothing to do with the familiar axial zero-field splitting parameter D, but it is a vector parameter, and the x means take the cross product (or vector product) an alternative way of writing is the determinant form... 

Lanthanide ions offer several salient properties that make them especially attractive as qubit candidates (i) their magnetic states provide proper definitions of the qubit basis (ii) they show reasonably long coherence times (iii) important qubit parameters, such as the energy gap AE and the Rabi frequency 2R, can be chemically tuned by the design of the lanthanide co-ordination shell and (iv) the same molecular structure can be realized with many different lanthanide ions (e.g. with or without nuclear spin), thus providing further versatility for the design of spin qubits or hybrid spin registers. 

J-splitting, when it exists, imposes the definition of new spin quantities. These quantities also evolve according to relaxation phenomena and may interfere (by relaxation) with the usual magnetization components. This latter interference stems precisely from cross-correlation rates, i.e., relaxation parameters which involve two different mechanisms, for instance the dipolar interaction and the so-called Chemical Shift Anisotropy (27,28) (csa)... 

This definition is analogous to that of the more common spin relaxation generalized order parameter, although the sensitivity to motional timescales is very different. As will be discussed later, a further distinguishing characteristic is that RDC analysis can in principle allow the explicit separation and determination of the direction and extent of the motional asymmetry. 

Another way to P takes into account the Fermi velocities of the up spin and down spin electrons as well. Since we use P as a phenomenogical fitting parameter, the detailed definition of P is not of concern here. 

The Q values in these tensors represent the Stokes parameters for the spin sensitivity of the detector in its x", y", z" frame. For example, Qx- describes the detector efficiency for measuring spin projections along +x" and —x", respectively (for the definition of the spin polarization vector see Section 9.2.1). 

Theoretical bases of continuum models including their mathematical formulation and numerical implementation have already been discussed in the previous chapter of this book. We have therefore restricted our review to the environment effects on the NMR observables, without going into the theory of continuum models. This contribution is divided into five sections. After the Introduction, the definitions of the NMR parameters are recalled in the second section. The third section is focused on methodological aspects of the calculation of the NMR parameters in continuum models. The fourth section reviews calculations of the solvent effects on the nuclear magnetic shielding constants and spin-spin coupling constants by means of continuum models, and the final section presents a survey on the perspectives of this field. 

In conclusion, field dependent single-crystal magnetization, specific-heat and neutron diffraction results are presented. They are compared with theoretical calculations based on the use of symmetry analysis and a phenomenological thermodynamic potential. For the description of the incommensurate magnetic structure of copper metaborate we introduced the modified Lifshits invariant for the case of two two-component order parameters. This invariant is the antisymmetric product of the different order parameters and their spatial derivatives. Our theory describes satisfactorily the main features of the behavior of the copper metaborate spin system under applied external magnetic field for the temperature range 2+20 K. The definition of the nature of the low-temperature magnetic state anomalies observed at temperatures near 1.8 K and 1 K requires further consideration. 

In the general case the proposed form of the wave function corresponds to the MP form but with matrices of infinite size. However, for special values of parameters of the model it can be reduced to the standard MP form. In particular, we consider a spin-1 ladder with nondegenerate antiferromagnetic ground state for which the ground state wave function is the MP one with 2x2 matrices. This model has some properties of ID AKLT model and reduces to it in definite limiting case. 

Here Ai are inverse mass parameters in WZ structure, corresponding to Luttinger parameters in ZB structure, me11 and me1 are k-dependent electron effective masses. D , aic and a2c are Bir-Pikus deformation potentials. Ai and 3A2,3 correspond to the crystal-field and spin-orbit splitting energies, respectively. The definition of several operators is given as L+ = (U iLy)/V2, a+ = (ax iay)/2,... 

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