Quantization axes

A rigorous quantum mechanical treatment of the combined interaction is laborious and computer-time consuming, because each interaction tends to have its own set of quantization axes, whereas the total perturbation does not quantize along either of the two sets of axes. 

Here, the magnetic perturbation is projected onto the quantization axes of Hq as the major interaction. 

We label the nuclear spin quantization axes (x",y",z") and require that the Sz> term has the form KSZJZ. 

Consider the matrix which transforms the I vector from the nuclear-spin into the electron-spin quantization axes ... 

Fig. 1.14. (A) Orientation of Ms and ML in the presence of internal molecular axes. (B) A case in which the external magnetic field determines the quantization axes.

Fig. 9. Spin quantization axes x, y, and z of the neighboring U4+(A)-U4+(B) exchange page in the U02 crystal lattice.

Obviously, different expressions are obtained for the alignment parameters, because these quantities are defined with respect to different quantization axes. However, it is possible to express the alignment properties in a unified manner, e.g., j/2oliin the coordinate frame with the quantization axis along the photon beam direction. Since the alignment tensor s/2k is defined in connection with statistical tensors, equ. (8.115c), one can use the rotation properties in equ. (8.82) to change the reference axis for the representation from the z-axis to the x-axis ... 

It appears that the three-space basis associated with each spin vector operator is arbitrary. Thus operator opS (and/or opI) need not necessarily be taken as quantized along some obvious physical direction, such as that of an applied magnetic field B. In other words, spin operators opS and opI need not be expressed in the same space, that is to be quantized along the same spatial directions (i.e. the spin projection quantum numbers may be measured along different selected directions in our three-space). The most general case, which occurs when the two quantization axes are not aligned, prevents the parameter matrices from being tensors. 

Much of the above is predicated on the major question, still to be completely and formally settled, as to whether, in quantum mechanics, specification of the quantization axis Zi for one spin leaves the quantization axes for all other spins in the system arbitrary, rather than compelling all of these to be identical to Zi. 

The rotor problem can be treated choosing different quantization axes. For near prolate tops the so-called F representation, c<->y is appropriate, while in the oblate case the choice z<- c... 

Accounting for spin-spin contributions to the line broadening is complicated by the different gyromagnetic ratio tensors of interacting particles, and the quantization axes of two spins may also be different. Assuming the Z, Z axes of the local coordinate systems... 

Under these conditions orientation of the rf field parallel to the field Hq (i.e., to the c-axis) can prove to be preferable. This is certainly so for H5A protons in effective quantizing fields which are almost perpendicular to the c-axis, if the applied fields are not too strong (see table 21). In this case the rf field will be equally oriented relative to the quantization axes of all the six proton groups with a given resonance frequency, and... 

The transition probabilities of the allowed (4) and forbidden (4) EPR transitions are given by Eq. (13), where 2rj is the angle between the nuclear quantization axes in the two manifolds with respect to Bq [7] ... 

Hamiltonian = matrix element of the Hamiltonian H I = nuclear spin I = nuclear spin operator /r( ), /m( ) = energy distributions of Mossbauer y-rays = Boltzmann constant k = wave vector L(E) = Lorentzian line M = mass of nucleus Ml = magnetic dipole transition m = spin projection onto the quantization axes = 1 — a — i/3 = the complex index of refraction p = vector of electric dipole moment P = probability of a nuclear transition = tensor of the electric quadrupole q = eZ = nuclear charge R = reflectivity = radius-vector of the pth proton = mean-square radi-S = electronic spin T = temperature v =... 

The quantization axes for electron and nuclear spin are fixed in the... 

By using four detectors in a standard Mott detector, a single gold foil can be used for spin analysis along two orthogonal directions. The polarimeter depicted schematically in Figure 3.2.2.32 combines two Mott detectors, one oriented at 90° with respect to the other, allowing for complete three-dimensional spin polarimetry, that is, with quantization axes =x, y, and z [39]. 

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