Spin operator rotation

An interesting variant of the resonance NSE is the so-called MIEZE technique [ 19]. Using two RF-flippers that operate at different frequencies a neutron beam is prepared such that a special correlation between a time varying spin rotation co= Qi-Q2) the velocity of the neutrons is achieved. An analyser after the second RF-flipper translates the spin rotation into an intensity modulation. The... 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

The 13C nuclei in formic and acetic acids relax faster and show larger nuclear Over-hauser enhancements than the 13C nuclei of the methyl esters (Table 3.18) [183], Hence a methyl ester is more mobile than its parent carboxylic acid, which is dimerized via hydrogen bonding. It may also be seen that a pure DD mechanism operates in formic acid, owing to the directly bonded proton (tjc = 2.0), whereas spin rotation also contributes to 13C relaxation in methyl formate (tjc = 1.55). 

Data System. An IBM-PC desktop computer is interfaced to the instrument control module to provide automated data collection and analysis. The data collection and analysis programs are menu driven. A data management facility is an integral part of the data system. A modeling utility is provided to aid the operator in chosing the operating conditions (rotational speed, spin fluid volume, fluid density, etc.) for a sample. Programming is done in compiled BASIC and the 8087 math coprocessor is used to improve computational speed. 

You will find that many of the sources do not use exactly the same matrix representations for some of the product operators and rotation matrices. The exact form of the density matrix depends on the numbering of the spin states and on certain conventions that are not consistent in the literature. In the above examples, the definitions are consistent with the product operator methods and with themselves. 

The magnetic moments given above will interact with an applied magnetic field, and these interactions are discussed extensively in chapter 8. In some diatomic molecules both nuclei have non-zero spin and an associated magnetic moment. The magnetic interactions which then occur are the nuclear spin-rotation interactions, represented by the operator... 

Fine structure terms spin-orbit, spin-spin and spin-rotation operators... 

In other words, each of the parameters is the sum of a first-order and a second-order contribution. We have met equation (7.126) for the effective rotational constant operator before, in an earlier section, where we pointed out that the second-order contribution Ba> is very much smaller than BiV) and that these two contributions have a different reducedmass dependence. It is importantto realise thatthis is not generally true. Indeed, except for molecules with very light atoms such as H2, the second-order contribution to the spin-rotation parameter is usually very much larger in magnitude than the first-order contribution. The same is also often true for the spin-spin coupling parameter /.. The reduced mass dependences of the two contributions to the spin-rotation parameter y are different from each other and quite complicated. However, Brown and Watson [17] were able to show the rather remarkable result that when one takes the first- and second-order contributions together as in equation (7.127), the reduced mass dependence of the resultant parameter y(R) is simply /u-1. 

The experimental evidence for such a contribution to the spin-rotation interaction in the effective Hamiltonian was somewhat elusive in the early days although there are now well documented cases of its involvement, for example for CH in its 4E state [21]. Equation (7.166) suggests one reason why this parameter is not as important in practice as might be expected. The last factor on the right-hand side of (7.166) is just the difference of the rotational constant operators for the upper and lower states. This causes a considerable degree of cancellation in a typical situation because the B value is not expected to vary markedly between the electronic states. 

Even though the spin orbitals obtained from (2.23) in general do not have the full symmetry of the Hamiltonian, they may have some symmetry properties. In order to study these Fukutome considered the transformation properties of solutions of (2.24) with respect to spin rotations and time reversal. Whatever spatial symmetry the system under consideration has, its Hamiltonian always commutes with these operators. As we will see, the effective one-electron Hamiltonian (2.25) in general only commutes with some of them, since it depends on these solutions themselves via the Fock-Dirac matrix. 

The advance of multidimensional NMR started from a product operator established by Ernst et al.4 and Kessler et al.123 The product operator is composed of matrices (operators of spin rotation correspond to the NMR pulse), which are combined by a product of matrix based on quantum mechanics with just a simple high-temperature approximation. The approximation quality is excellent. This is a great advantage of the NMR method compared to other spectroscopic methods, such as the Raman spectrum. 

Interaction between the electron spin and the rotational angular momenta of the nuclei, HSR = spin-rotation operator. 

This operator accounts for the interaction between the electron spins and the magnetic field created by nuclear motion. As the nuclei are heavy, their angular velocity is approximately m/M times smaller than the angular velocity of the electrons. Consequently, except for light molecules, the spin-rotation interaction is very small compared to the spin-orbit interaction. The microscopic Hamiltonian from Kayama and Baird (1967) and Green and Zare (1977) has the case (a) form,... 

Second-order effects arising from the product of matrix elements involving J+ L and L+ S operators have the same form as 7J+S. In the case of H2, the second-order effect seems to be smaller than the first-order effect, but in other molecules this second-order effect will be more important than the first-order contribution to the spin-rotation constant. These second-order contributions can be shown to increase in proportion with spin-orbit effects, namely roughly as Z2, but the direct spin-rotation interaction is proportional to the rotational constant. For 2n states, 7 is strongly correlated with Ap, the spin-orbit centrifugal distortion constant [see definition, Eq. (5.6.6)], and direct evaluation from experimental data is difficult. On the other hand, the main second-order contribution to 7 is often due to a neighboring 2E+ state. Table 3.7 compares calculated with deperturbed values of 7 7eff of a 2II state may be deperturbed with respect to 2E+ by... 

There is a large class of relaxation mechanisms which operate on molecules in motion in non-metailic samples. All but one, the spin-rotation interaction, depend on the fact that the change in the molecular orientation or translation modulates the field due to that particular interaction and creates a randomly varying field at the site of the nucleus in question. Any such random motion can have associated with it a special form of an autocorrelation function G(t), expressed in terms of a scalar product of the local field h(t) and the same local field at an earlier time h(0), which is a measure of... 

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