Lineshape transition probability

In order to define the transitions in fig. 4, we need to go back and examine the transition probability, which is usually given by time-dependent perturbation theory [36]. If (j>i is the initial state, 0f the final state and Ix the perturbation, then the transition probability is given in eq. (4). Two factors make up the transition probability, and they are complex conjugates, so the intensity is real. In the generalization presented here, these two factors have a physical interpretation they are the projections of the individual transitions along the total magnetization. In a dynamic system, the two factors are not complex conjugates, so the lineshapes in fig. 4 are more complicated. In spite of this, we may still treat them as transitions, as in the static case. 

In an analogous fashion the numerically obtained transition probabilities shown in Fig. 14.6 can be converted to cross sections by integrating over impact parameter and averaging over the possible collision velocities. Collisions with v 1 E do not have a single allowed value of 6. However, since the lineshape of Fig. 14.6(b) is simple, some averaging over the possible values of 6 has no appreciable effect. For v E, 6 and 0 are fixed, at 90° and 0°, so the integration over impact parameter is only over b, and the calculated v E cross section is very similar to Fig. 14.6. 

Fig. 15.11 The K 29s + 27d resonance in the presence of a low frequency rf field. In zero rf field (a), the FWHM is 1.6 MHz. In (b)-(d), a 1.0 MHz field of strength 0.05 V/cm, O.lV/cm, and 0.2 V/cm respectively is present. The solid line in (b) is a numerical integration of the transition probability, and the bold line is the convolution of a Lorentzian lineshape with a sinusoidal shift from resonance. In (e), the rf frequency is 0.5 MHz and its strength is 0.2 V/cm. For these low frequencies, the features are no long frequency dependent but rather are field strength dependent (from ref. 18).

In this formula, I is the nuclear quantum number, r(Bj) the first derivative lineshape function, B the resonance position and P the transition probability. 0 and i are the Euler angles expressing the orientation of the magnetic field vector B with respect to the principal axes of the tensors. Integration is needed since in powder samples, the crystallites take all possible orientations with respect to the magnetic field. Since the principal tensor axes and the crystal axes are assumed to be coincident, integration can be restricted to one octant of the unit sphere. 

One of the goals of the calculations that we will be describing in later sections is to obtain the lineshape function, the survival probability and transition probabilities asso-... 

RRGM summary. In this section, we have reviewed the recursive residue generation method for the computation of the lineshape function and the transition probabilities. The RRGM aims at computation of the set of residues and eigenvalues ra, Ea. The steps are summarized as follows ... 

W.L. Wiese Transition probabilities . In Methods of Experimental Physics, Vol. 7a, ed. by B. Bederson, W.L. Fite (Academic, New York 1968) p. 117 W.L. Wiese, M.W. Smith, B.M. Glennon Atomic Transition Probabilities. Nat l Standard Reference Data Series NBS4 and NSRDS-NBS22 (1966-1969), see Data Center on Atomic Transition Probabilities and Lineshapes, NIST Home-page (www.nist.org) ... 

Experimentally the continuous-wave (CW) EPR experiment is a field-swept experiment in which the microwave frequency (Vc) is held constant and die magnetic field varied. Computer simulations performed in field space assume a symmetric lineshape function,/in Eq. (3) (J(B - B es), CTg), which must be multiplied by dv/dB and assume a constant transition probability across a given resonance [1,29]. Sinclair and Pilbrow [30,31] have described the limitations of fliis approach in relation to asymmetric lineshapes observed in high-spin Cr(III) spectra and the pres-... 

Needless to say, the so-called combination transitions are also considered in this subspace. Lineshape equations for special forms of the relaxation matrix can also be written in terms of the Hilbert space. However, the notation becomes quite involved. This is probably the source of some erroneous simplifications which consist of neglecting combination transitions in the equations of lineshape. (50)... 

It is important to understand that the origin of the different frequency shifts experienced by different molecules is the same stochastic frequency modulation <5stochastic variable, which persists on the timescale of the measurement. In this limit the observed lineshape is determined not by the dynamics of m(z) but by the probability P u) ) that at any time the molecule is characterized by the instantaneous transition frequency coT If the normalized absorption profile of an individual molecule is given by a a> — co ), where a co) peaks at m = 0 and dcoalco) = 1, the observed lineshape is... 

According to the above discussion, Pi/2 and v, involve the transitions of proton and C nuclear spins. This supports the above interpretation that these transitions are related to the interaciton of proton and C nuclear spins because the pristine system also contains nuclei with a natural abundance of 1.1%. On the other hand, the second harmonic signal, 2v which has a comparable intensity to those of Pi/2 and Vpiv e in the C-enriched system in Figure 6.24, completely lacks it in the pristine system, as seen in Figure 6.25. This is reasonably expected because the probability of a C — C pair in the pristine system is only about 1% of that of a C—H pair. From these results it can be concluded that the C-enriched system is not adequate for observing the soliton spin density because the lineshape distortion is caused by the overlap of the structures related to multiple and fractional nuclear quantum transitions. Further studies of the origin of these transitions would be interesting. 

Figure 3. The XSophe (v 1.1.4) main Window. The interface allows creation and execution of multiple input files on local or remote hosts. There are macro task buttons to guide the novice through the various menus and two button bars to allow easy access to the menus. For example, the bottom bar (left to right), Experimental Parameters, Spin System, Spin Hamiltonian, Instrumental Parameters, Single Crystal Settings, Lineshape Parameters, Transition Labels/Probabilities, File Parameters, Sophe Grid Parameters, Optimisation Parameters, Execution Parameters and Batch Parameters.

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