Sphericity operational

As you can see, the bookkeeping for MQCs gets pretty messy, and we will see later (Chapter 10, Section 10.4) that another kind of operator called spherical operators is neater and easier to visualize for MQCs. 

EXPANDING OUR VIEW OF COHERENCE QUANTUM MECHANICS AND SPHERICAL OPERATORS... 

The effect of the spherical operators on individual spin states is actually opposite to this for example 1+ j8 > -> la >. It is the magnetic quantum number that is raised by the operator —1/2 (j8 state) to +1/2 (a state). In this book, we will reverse the definition for convenience so that the operators make intuitive sense I" " raises the spin state from a to jS. 

The most important thing about the raising and lowering (or spherical ) operators is the way they react to gradients, which is to say their coherence order. The coherence order is no longer ambiguous. For the heteronuclear system... 

The coherence order p can be understood precisely if we consider the effect of a gradient on one of the spherical operators. A gradient is just like chemical-shift evolution except that the amount of evolution depends on the position in the tube (z coordinate) rather than the chemical shift. 

One more thing we can do with spherical operators We can easily derive the expressions given in Chapter 8 for pure ZQC and DQC. Start with the spherical product I+S+ and... 

The Cartesian operators have mixed coherence order because they are linear combinations of the spherical operators... 

The effect of gradients is especially simple to understand if we consider spherical operators. A Cartesian operator, such as Ix or ly, will precess during a gradient according to... 

This is the position-dependent phase shift expressed in terms of the x and y components of the magnetization. Here, we assume that the I spin chemical shift is on-resonance, so that ordinary chemical-shift evolution can be ignored during the gradient. Now we can look at the effect of a gradient on the spherical operators by expressing them in terms of the Cartesian operators ... 

In terms of spherical operators, the DQF-COSY experiment looks like this... 

The 180° pulses in the center of the spin echoes invert the spherical operators, with I" " becoming I- and I- becoming I" ". We have to take this into account when designing gradients now pi = — 1 (I Ig) and p2 = +2(1+1+), so we need gradient strengths of G = 2 and G2 = 1 to select this coherence pathway. 

To use the more formal analysis of phase cycling developed in Section 10.6, we first need to describe the coherence pathway in terms of spherical operators (I+, S, etc). Starting at the end and working backward and using the convention of positive coherence order during... 

Taking into account all the relevant criteria spin-1/2 nuclei in the liquid phase can generally be described using CARTESIAN, spherical and shift product operators as shown in Table 2.3. The spherical operators are not shown because they can be easily derived from the shift operators, see Table 2.5. 

By convention a pure absorption signal using the quadrature detection procedure is composed of a real part Ik,y and a imaginary part Ik,x- Conversion to spherical operators means that i Ik, i = (Ik,y + i Ik,xV 2 represents a pure absorption signal, while Ik,+i represents a pure dispersion signal. The operator Ik,+i corresponds to the "quad image". 

These two invariants are at the origin of two spherical operators one corresponds to the constant scalar 0Aig, and the other to 4Aig. The former invariant may be obtained by taking the norm of the entire (/-manifold, which can be expressed in the Aig functions of Eq. (7.16) as follows ... 

In the theory of crystal fields the following spherical operators are often used ... 

The numerical factors Op given in table 4 couple spherical operators to Stevens polynomials ... 

Proportionality factors between spherical operators and uniform spherical polynomials... 

Crystal-field parameters other parameters for spherical operators... 

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