Spinning matrix

The only approximation to be admitted at this stage will be that inherent in the separability anaatz (1) with the constraint of strong orthogonality. In this case there is a corresponding separability of the density functions, embodied in two theorems [1,2] for a separable system, comprising subsystems A, B,. . R,., , the one-body density matrix (spin included) takes the form... 

ESR techniques use nitroxyl radicals either dispersed in polymer matrix (spin probe) or covalently bonded to polymer chain (spin label) which are sensitive to the environment allowing molecular motion and microstructure of polymers to be identified from spectra (103). Quantitative methods of heterogeneous ESR spectra are divided into (J) outer hyperfine etrema, (2) signal intensities related to the relative concentration of the probe in different phases, and (3) simulation of the spectra. The presence of two well-separated outer maxima above the glass-transition temperature could be ascribed to two phases in natural rubber (104), miscible blends (105), immiscible blends (106), cross-linked polymers (107), and polyurethanes (108). ESR has used the measurement of the oxidation product to monitor the consumption of stabilizer in polypropylene (109). 

In 1946, Bloch [ 16] presented a theory for nuclear spin relaxation in which he derived a set of equations of motion to predict the behaviour of an ensemble of isolated spins interacting weakly with the lattice. In brief, the Bloch equations describe the evolution of the longitudinal (diagonal elements of the density matrix) and transverse (off-diagonal elements of the density matrix) spin magnetisation to their respective equilibrium values phenomenologically, with the first order rate of both processes... 

Finally, we consider the complete molecular Hamiltonian which contains not only temis depending on the electron spin, but also temis depending on the nuclear spin / (see chapter 7 of [1]). This Hamiltonian conmiutes with the components of Pgiven in (equation Al.4,1). The diagonalization of the matrix representation of the complete molecular Hamiltonian proceeds as described in section Al.4,1.1. The theory of rotational synnnetry is an extensive subject and we have only scratched the surface here. A relatively new book, which is concemed with molecules, is by Zare [6] (see [7] for the solutions to all the problems in [6] and a list of the errors). This book describes, for example, the method for obtaining the fimctioiis ... 

Binsch [6] provided the standard way of calculating these lineshapes in the frequency domain, and implemented it in the program DNMR3 [7], Fonnally, it is the same as the matrix description given in section (B2.4.2.3). The calculation of the matrices L, R and K is more complex for a coupled spin system, but that should not interfere witii the understanding of how the method works. This work will be discussed later, but first the time-domain approach will be developed. 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

Relaxation or chemical exchange can be easily added in Liouville space, by including a Redfield matrix, R, for relaxation, or a kinetic matrix, K, to describe exchange. The equation of motion for a general spin system becomes equation (B2.4.28). 

A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. 

In the usual preparatioii-evohition-detection paradigm, neither the preparation nor the detection depend on the details of the Hamiltonian, except hi special cases. Starthig from equilibrium, a hard pulse gives a density matrix that is just proportional to F. The detector picks up only the unweighted sum of the spin operators,... 

For example, the observed transitions of an AB spin system have a Liouville matrix given m equation (B2.4.35). The coupling constant is J, and it is assumed that ciig = = -5/2, so that 5 is the frequency... 

The xy magnetizations can also be complicated. Eor n weakly coupled spins, there can be n 2" lines in the spectrum and a strongly coupled spin system can have up to (2n )/((n-l) (n+l) ) transitions. Because of small couplings, and because some lines are weak combination lines, it is rare to be able to observe all possible lines. It is important to maintain the distinction between mathematical and practical relationships for the density matrix elements. 

The so-ealled Slater-Condon rules express the matrix elements of any one-eleetron (F) plus two-eleetron (G) additive operator between pairs of antisymmetrized spin-orbital produets that have been arranged (by permuting spin-orbital ordering) to be in so-ealled maximal eoineidenee. Onee in this order, the matrix elements between two sueh Slater determinants (labelled >and are summarized as follows ... 

By applying Eq. (C.13) to the spin operators Si and using Eq. (C.22), one then gets after some matrix multiplications... 

The fact that there is a one-to-one relation between the (—1) terms in the diagonal of the topological matrix and the fact that the eigenfunctions flip sign along closed contours (see discussion at the end of Section IV.A) hints at the possibility that these sign flips are related to a kind of a spin quantum number and in particular to its magnetic components. 

The two sets of coeflicien ts, one for spin-up alpha electrons and the other for spin-down beta electrons, are solutions of iw O coupled matrix eigenvalue problems ... 

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