Lineshape equations

One important relationship which has survived from the earliest days of DNMR [1] and which is still used to estimate exchange rates is the relationship between the exchange rate which is the rate that just results in coalescence of the A and B signals into a broad, single, flat-topped absorption. From differentiation of the lineshape equation, it is possible to obtain the simple expression [1]... 

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)... 

One should note that in reports on intermolecular spin exchange there are often errors in the renormalization of the vectors in composite Liouville space. (14, 51, 78) Such errors lead to erroneous lineshape equations in cases where the numbers of nuclei in the components of an equilibrium are not equal. 

Recently an effective method of calculating the absorption mode of a dynamic NMR spectrum by the point by point approach has been suggested. (47) It is based on the symmetric form of the lineshape equation (147). One can always choose a real unitary matrix U such that only the last element of the vector F = Uf is non-zero. The matrix ought to have its last column equal to the vector (fff ) 1/2f. Upon a transformation of equation (149), with the matrix U, one obtains the /abs ([Pg.261]

For exchange between two equally populated sites (as with the methyl groups in DMF Section 10.2), the lineshape equation [Eq. (10.1)] that describes NMR signal intensity [/(v)] at each point along the frequency (v) axis2 is ... 

Here Mi and Ma are the second and the fourth moment of the resonance lineshape. Equation (30) defines the exchange field Hg. In the well-known relation, AHg = cHo/He, where Hd defines the dipolar broadening, a factor c emerges. Van Vleck (1948) calculated this factor for the two extreme cases H H (c = 1) and Hz 10/3). Hence, the... 

Fortunately, for non-integer quadnipolar nuclei for the central transition = 0 and the dominant perturbation is second order only (equation Bl.12.8) which gives a characteristic lineshape (figure B1.12.1(cB for axial synnnetry) ... 

Application of tiiis approach to equation (B2.4.37) gives equation (B2.4.40). If = -co = -S/2, the synnnetry of the matrix and one additional transfomiation means that it can be broken into two 2x2 complex matrices, which can be diagonalized analytically. The resulting lineshapes match the published solutions [13]. 

Two different approaches have been followed to calculate the lineshapes within a relaxation model. According to a phenomenological approach based on the modified Bloch equations [154, 155], the intensity distribution of the theoretical Mossbauer spectrum may be written as [156] ... 

Equation (51) has a clear physical interpretation. Recalling the lineshape for a single excitation route, where fragmentation takes place both directly and via an isolated resonance [68], p oc (e + q)2/( 1 + e2), we have that 8j3 is maximized at the energy where interference of the direct and resonance-mediated routes is most constructive, e = (q I c(S )j2. In the limit of a symmetric resonance, where q —> oo, Eq. (51) vanishes, in accord with Eq. (53) and indeed with physical intuition. The numerator of Eq. (51) ensures that 8]3 has the correct antisymmetry with respect to interchange of 1 and 3 and that it vanishes in the case that both direct and resonance-mediated amplitudes are equal for the one-and three-photon processes. At large detunings, e —> oo, and 8j3 of Eq. (51) approaches zero. 

Implementation of Equation 9.18 in spectral simulators requires some extra precautions (Hagen 1981 Hagen et al. 1985d) (A) The increased periodicity now requires one half of the unit sphere to be scanned. (B) The fact that the term within the absolute-value bars in Equation 9.18 can change sign as a function of molecular orientation implies the possibility that for specific orientations the linewidth becomes equal to zero. To avoid a program crash due to a zero divide, e.g., in the expression for the lineshape in Equation 4.8, a residual linewidth W0 has to be introduced ... 

In the full quantum mechanical approach [8], one uses Eq. (22) and considers both the slow and fast mode obeying quantum mechanics. Then, one obtains within the adiabatic approximation the starting equations involving effective Hamiltonians characterizing the slow mode that are at the basis of all principal quantum approaches of the spectral density of weak H bonds [7,24,25,32,33,58, 61,87,91]. It has been shown recently [57] that, for weak H bonds and within direct damping, the theoretical lineshape avoiding the adiabatic approximation, obtained directly from Hamiltonian (22), is the same as that obtained from the RR spectral density (involving adiabatic approximation). 

Using the case of S = 5/2 as an illustrative example, he demonstrated that it was possible to derive closed-form analytical expressions for the PRE of the form of the SBM equations times (1 + correction term). For typical parameter values, the effect of the correction term was to increase the prediction of the SBM theory by 5-7%. A similar approach was also applied to the S = 7/2 system, such as Gd(III) (101), where the correction terms could be larger. For that case, the estimations of the electron spin relaxations rates, obtained in the solution for PRE, were also used for simulations of ESR lineshapes. 

Various theoretical formalisms have been used to describe chemical exchange lineshapes. The earliest descriptions involved an extension of the Bloch equations to include the effects of exchange [1, 2, 12]. The Bloch equations formalism can be modified to include multi-site cases, and the effects of first-order scalar coupling [3, 13, 24]. As chemical exchange is merely a special case of general relaxation theories, it may be compre-... 

The quantities U Ix)j and UFx)j in (13) are projections of the eigenvector j along lx- From the above equations, we can interpret these as follows. The term UFx)j is the amount that the transition j received from the total X magnetization, created from the equilibrium state, and (U Ix)j is how much that transition contributes to the observed signal. These two terms may not be equal, as we see in exchanging systems. This general approach forms the basis of the description of dynamic NMR lineshapes. 

Figure 1 Simulations of CT lineshapes corresponding to (A) static and (B) MAS experiments, for different values of the asymmetry parameter rjq.The positions of the isotropic chemical shift (c5iso) and some well-defined singularities are shown, in terms of the parameter A, defined in Equation 8. All simulations were done with the DMFIT software. ...

Additional evidence for the applicability of the above data interpretation is obtained by using Equation 3 to predict values of l/(Tlp) min. These are 2.8 X 104 sec-1 and 6.4 X 103 sec-1 for H = 4.7 G and 20.6 G, respectively, and are consistent with the T p and T2 data. The motional properties reported here differ significantly from previous measurements the activation energy obtained from a temperature above 120°C is two to three times the values obtained in Ref. 13, and motional narrowing of the lineshape occurs at a significantly higher temperature than that in Ref. 12. 

We shall shortly consider such fundamental concepts as density matrices and the superoperator formalism which are convenient to use in a formulation of the lineshape theory of NMR spectra. The general equation of motion for the density matrix of a non-exchanging spin system is formulated in the laboratory (non-rotating) reference frame. The lineshape of a steady-state, unsaturated spectrum is given as the Fourier transform of the free induction decay after a strong radiofrequency pulse. The equations provide a starting point for the formulation of the theory of dynamic NMR spectra presented in Section III. The reader who may be interested in a more detailed consideration of the problems is referred to the fundamental works of Abragam and... 

---