Lineshapes rigid-limit

Similar to fluorescence depolarization and NMR, two limiting cases exist in which the molecular motion becomes too slow or too fast to further effect the ESR lineshape (Fig. 8) (35). At the fast motion limit, one can observe a narrow triplet centered around the average g value igxx + gyy + giz with a distance between lines of aiso = Axx- -Ayy- -A2,z)l3, where gu and Ajj are principal values of the g-tensor and the hyperflne splitting tensor A, respectively. At the slow motion limit, which is also referred to as the rigid limit, the spectrum (shown in Fig. 8) is a simple superposition of spectra for all possible spatial orientations of the nitroxide with no evidence of any motional effects. Between these limits, the analysis of the ESR lineshape and spectral simulations, which are based on the Stochastic Liouville Equation, provide ample information on lipid/protein dynamics and ordering in the membrane (36). 

Fig. 6 Structural details obtained by lineshape analysis, (a) In KcsA, labeling at any residue position renders a tetramer with potentially four spin labels, (i) Tandem dimer construct (ii) with cys residues in both protomers (control used to evaluate the effects of the intersubunit linker) and (iii) with only one of the protomers containing a cys (used in the analysis), (b) Rigid-limit X-band EPR spectra obtained at pH 7 (thick line, closed state) and at pH 4 (thin line, open state). Right panel, absorption spectra obtained from integration and relative fits obtained with convolution superimposed, (c) Simulated spin- and amplitude-normalized spectra for the two interspin distances in the figure (100% spin labeling efficiency), (d) Helical wheel representation of residues 100-119. Both closed (top) and open (bottom) states are represented as pairs of helical wheel...

This is demonstrated in Fig. 10. The calculated lineshapes refer to CSL probes (see Fig. 5), undergoing rotational diffusion in an isotropic medium. Drastic spectral changes are observed when the motion increases from the rigid limit (a) to the fast-rotational region (e). 

Figure 3. ESR lineshapes of spin probes in frozen (a) and mobile (b) hosts, (s The magnetic dipoles m of immobile spin probes in a frozen liquid have different u>o values due to their different orientations with respect to Bo, thus resulting in a broad line with width Arno (black line), usually referred to as rigid-limit or pwder lineshape. (b) If the spin probe undergoes rotation (sketched as instantaneous clockwise jumps at random times), ft>o fluctuates. When the rotational rate 1/r is larger than the width of the coo distribution Acoo, the different precession frequencies become indistinguishable and an average value is seen, that is, the ESR lineshape coalesces (motional narrowing) [4,14]. Adapted from Ref. [26].

As explained above, some approximations have been made. Firstly, a linear limit has been assumed, which implies that both Zeeman splittings of A and / are small compared to kT (see Eq. (175) and the bandwidth (see Eq. (178)). Another approximation, also designated as the "rigid shift approximation", assumes that the magnetic field influence on the lineshape can be neglected so that only a "rigid shift" is taken into consideration (Piepho and Schatz, 1983). 

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