Hyperfine relative intensities

Most Mossbauer spectra are split because of the hyperfine interaction of the absorber (or source) nuclei with their electron shell and chemical environment which lifts the degeneracy of the nuclear states. If the hyperfine interaction is static with respect to the nuclear lifetime, the Mossbauer spectrum is a superposition of separate lines (i), according to the number of possible transitions. Each line has its own effective thickness t i), which is a fraction of the total thickness, determined by the relative intensity W of the lines, such that t i) = Wit. 

From (4.56) and Table 4.3, we derive the relative intensity ratios 3 2 1 1 2 3 for the hyperfine components of a Zeeman pattern of a powder sample. The transition probability for the case of the polar angle 6 = Oq can readly be calculated by integrating (4.56) only over the azimuthal angle (j). One obtains a factor (1 + cos 0o)/2 and sin 0o for m = 1 and m = 0, respectively, which are multiplied by the square of the Clebsch-Gordan coefficients. As a consequence of the angular correlation of the transition probabilities the second and fifth hyperfine components (Fig. 4.17) disappear if the direction k of the y-rays and the magnetic field H are parallel (0q = 0). 

Effect of Crystal Anisotropy on the Relative Intensities of Hyperfine Splitting Components... 

Fig. 7.53 Transmission Mossbauer spectra of the 137, 155, and 187 keV nuclear transitions of 186,188,190qj taken with sources emitting an unsplit line and (a) Os02-absorber (rj 0), (b) OSP2-absorber (rj 0.74). The curves are the results of least-squares fits. The vertical bars indicate the positions and relative intensities of the individual hyperfine components (from [254])...

Things get a little more complicated when a spin 1 nucleus like 14N is added to the picture, but the same technique works again for the determination of the relative intensities of the ESR lines. Consider, for example, the relative intensities of the hyperfine lines arising from the pyrazine anion radical, whose spectrum is shown in Figure 2.3. Like that of the naphthalene anion radical, the spectrum observed for the pyrazine anion radical2 consists of 25 well-resolved... 

The (hyperfine) interaction between magnetic nuclei and unpaired electrons causes characteristic signal patterns the spacing and relative intensities of the signals... 

Further information with respect to ZnS Mn nanocomposite phosphors is available from EPR spectra. The ZnS Mn/AA nanocrystal shows a typical sextet in X-band (9-GHz) ESR spectra, as shown by curve (a) in Figure 13.2.5. In contrast, the sample without AA exhibits the second component, signal II (Fig. 13.2.5b). From the sample with sulfur deficiency, ZnS(0.8) Mn, the relative intensity of signal II is higher (Fig. 13.2.5c). On the other hand, the g value, 2.0013, is significantly different from that of interstitial Mn(II) (g = 2.020) (19). From this, together with the value of the hyperfine structure constant (13), it is reasonable to assume that signal II is associated with Mn(II) with its coordination number lower than 3. 

Fig. 20. Dependence on temperature of the resolved hyperfine-shifted resonances of ferricytochrome c. The numbers on the right give the relative intensities of the lines...

The number of isotropic hyperfine lines from a particular nucleus depends on the nuclear spin, I, and the line multiplicity is 2/ + 1. For n equivalent nuclei, the EPR spectrum consists of 2nl + 1 lines whose relative intensities are given by binomial coefficients obtained in the expansion of (1 + x)n (Knowles et al., 1976). When nuclear hyperline interactions occur, Eq. (16.4) becomes... 

A Mossbauer spectrum contains different absorption peaks. The relative intensity, position, and shape of the absorption peaks composing the spectrum are related to the hyperfine interactions affecting the sample nuclei. The 57Co nuclei contained in the source are included in a solid matrix, such as a face-centered cubic (FCC) rhodium, which only affects the position of the emission line but does not split it [140],... 

EPR spectrum of electrochemically generated benzene radical anion, C6H6-. The hyperfine interaction between the free spin and the six H1 nuclei generates a seven-line spectrum of nominal relative intensities 1 6 15 20 15 6 1. The hyperfine splitting constant is 0.375 mT. 

It is, of course, also necessary to calculate the relative intensities of the hyperfine components of each rotational transition in order to assign the spectrum. As we have seen elsewhere, the perturbation due to the interaction of the microwave electric field E(t) with the molecular electric dipole moment may be represented by the effective Hamiltonian... 

Table 5.5.2. Binomial coefficients for the relative intensities of hyperfine lines (spin = 1/2)...

In the present experiment, we are concerned with the hyperfine structure of the ben-zosemiquinone radical anions. The delocalized unpaired tt electron is of course distributed over the entire molecular frame of six C atoms and two O atoms. With R = H, by symmetry, it is clear that the four protons are all equivalent in the para species hence five hyperfine lines with relative intensities 1 4 6 4 1 are expected in the ESR spectrum of this radical. By contrast, when R is not a proton, the three ring protons are not related by symmetry, and thus each may be expected to possess a different splitting constant. A hyperfine structure pattern of eight unequally spaced lines of equal intensity is expected. The line splittings and relative intensities in ESR spectra thus convey information about the geometric arrangement of the atoms. 

Regarding the relative intensities of the observed spin polarization mechanisms, it is also important to note that the [3-hyperfine interactions in these radicals are conformationally modulated, and this process can also quench RPM polarization. In a qualitative way, we can consider the modulation process to be a relaxation mechanism that exchanges magnetization between different nuclear spin orientations. Since these different orientations can have opposite phases of RPM polarization, the exchange of emissive and absorptive lines can cancel the intensity of the transitions. 

More sensitive ESR measurements over a wider magnetic field range find additional resonances in phosphorus- and arsenic-doped material, examples of which are shown in Fig. 5.11. The extra lines (two for phosphorus and four for arsenic) are due to the hyperfine interaction of the electron bound to the donor (Stutzmann and Street 1985). The ESR spectra have exactly the number of lines and relative intensities expected from the nuclear spins of and for phosphorus and arsenic atoms. The splitting of the lines, is proportional to the electron density at the nucleus and is a measure of the localization length, r, of the donor. 

Above 100 K, motional effects on spectrum become pronounced with increasing temperature and, above 230 K, the spectra consist of essentially an isotropic and equally spaced hyperfine triplet, but with different relative intensities. The line shape simulations were carried out by adopting a Brownian rotational diffusion model in order to evaluate the associated (average) rotational correlation time, and its degree of anisotropy, = zpy, /... 

There is normally interaction with more than one magnetic nucleus. Consider, for example, the methyl radical. Each proton may be in either of two spin states, a and jS, so that there are four possible resultant magnetic spin-states ,a,a ,a,jS a,j8,j8 j8,j8,)3. The spectrum therefore has four components, and since the configurations ,a,j8 and a,j3,j8 may each be achieved in three ways whereas the other two configurations are unique, the relative intensities of the absorptions are 1 3 3 1. The resulting spectrum is shown in Fig. 1 the radical was generated by the interaction of t-butyl hydroperoxide with titanous ion (Section IID) and the hyperfine splitting constant, —the separation of the mid-points of... 

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