Interaction frame

Prior to an effective Hamiltonian analysis it is, in order to get this converging to the lowest orders, typical to remove the dominant rf irradiation from the description by transforming the internal Hamiltonian into the interaction frame of the rf irradiation. This procedure is well established and also used in the most simple description of NMR experiments by transforming the Hamiltonian into the rotating frame of the Zeeman interaction (the so-called Zeeman interaction frame). In the Zeeman interaction frame the time-modulations of the rf terms are removed and the internal Hamiltonian is truncated to form the secular high-field approximated Hamiltonian - all facilitating solution of the Liouville-von-Neumann equation in (1) and (2). The transformation into the rf interaction frame is given by... 

It applies for both formulations above that the expansion in principle contains an infinite number of terms. The convergence to a few lowest order terms relies on the ability to orderly separate influences of the dominant rf irradiation terms (through a suitable interaction frame) from the less dominant internal terms of the Hamiltonian. In principle, this may be overcome using the spectral theorem (or the Caley-Hamilton theorem [57]) providing a closed (i.e., exact) solution to the Baker-Campbell-Hausdorf problem with all dependencies included in n terms where n designates the dimension of the Hilbert-space matrix representation (e.g., 2 for a single spin-1/2, 4 for a two-spin-1/2 system) [58, 59]. 

Dipolar recoupling may also be accomplished using continuous rf irradiation as demonstrated in the heteronuclear and homonuclear case by the CP at MAS conditions (or for low-y heteronuclear spins called double-cross-polarization, DCP [103]) and HORROR (homonuclear rotary resonance) [26] experiments, respectively. These experiments may easily be described by transforming the description into the interaction frame of the rf irradiation using (14a) exploiting... 

Upon entering the interaction frame of the rf irradiation for the CNvn or RN n sequences [cf. (14)] and taking the first-order effective Hamiltonians [cf. (17a) and (18a)], it is possible to establish the following selection rules for the averaging (and conversely recoupling) of the various interactions described in (45) as... 

In the simplest setup, the two strong field components may be set identical to Ci = Cs = C. The relatively large CIX or CSX term averages isotropic and anisotropic chemical shift effects as well as the heteronuclear dipolar coupling interaction between 15N or 13C and H. The difference of - or the sum of - the B coefficients selects the form of the recoupled heteronuclear dipole-dipole coupling interaction, as expressed in terms of the effective Hamiltonian in the interaction frame of the rf irradiation... 

MHz, a it/2 pulse was 2.2 ysec and the data were acquired with the carrier frequency below resonance (the rhs of the spectrum in each figure). These spectra may be used to determine polymer crystallinity and to determine various kinds of macromole-cular motion. Also spin-lattice relaxation times in the rf interaction frame (Ti ) addition to conventional T- and T p relaxation times have been measured to help elucidate the various mechanisms responsible for the observed chemical shift line shapes. 

In the cyclic case it is most convenient to transform the spin system to the RF interaction frame, defined by the transformation operator URp t). The total Hamiltonian in the RF interaction frame becomes [26]... 

The fact that we can express the propagator in this form allows us also to expand the reduced density operator in the RF interaction frame as... 

The form of the RF interaction Hamiltonian in Fourier space in Eq. 23a suggests that we can transform it to a space where it becomes time independent. Transforming the Fourier density operator to a new interaction frame defined by... 

In this Appendix the k values of the non-zero Fourier series coefficients of in the RF interaction frame for the CN and RN sequences are derived. 

When a tt pulse segment with an overall phase of = 0 applied during a time interval 0 < f < Tc/N is represented in the RF interaction frame by the transformation... 

Fig. 2. Schematic K X-ray spectra induced by 1.4 MeV/amu H and He impact. Here white lines indicate observed spectra [9] and black lines spectra calculated in the direct Coulomb interaction frame.

We now calculate the perturbation to the Zeeman field due to the quadrupolar interaction by means of average Hamiltonian theory.This is accomplished by transforming TYq to the Zeeman interaction frame and then applying the spherical tensor rotation properties to the spin elements 72,The resulting quadrupolar Hamiltonian TTq in the rotating frame is given by ... 

As usual, polarization-transfer experiments where the relevant quantization axis is the static magnetic field will be called laboratory-frame experiments, while experiments where the applied r.f. field provides the relevant quantization will be referred to as rotating-frame experiments. This classification should not be confused with the frame of reference where the Hamiltonian is set up and the equation of the motion is solved in. Usually, laboratory frame polarization-transfer experiments are described in a rotating (or doubly rotating in heteronuclear cases) frame of reference that is an interaction frame with respect to the dominant Zeeman interaction. Rotating-frame experiments are usually described in a frame of reference obtained by going into a further interaction representation, this time with respect to the r.f. field. 

The intensities I can be identified with the intensities of the nth spinning side bands of the chemical-shielding difference tensor [74]. In the time-dependent interaction frame the Hamiltonian is given by... 

We take an isolated heteronuclear spin pair (S , S2), where Si represents the observed nucleus, and rotation-synchronized tt pulses are applied at the Si and S2 Larmor frequencies. In the Zeeman interaction frame, the Hamiltonian... 

After transformation to a time-dependent interaction frame defined by... 

To make the series expansion in Eq. (4) converge faster, it is usefiil to transform the description into an interaction frame, where the Hamiltonian is divided into a larger and a smaller part, e.g., H t) =Ho t) + Hi t) with Hi (t) being the larger term. With this approach, we can separate out the evolution due to the larger part of the Hamiltonian by writing the spin propagator as a product [29]... 

---