Relative intensities of absorption lines

The interpretation of a complex Mossbauer spectrum will obviously be simplified if the relative intensities of the various components are known. Once the energy levels of the Zeeman/quadrupole Hamiltonian have been calculated, and the spin quantum numbers for each state assigned (or appropriate linear combinations if the states are mixed), it is possible to calculate the intensities from the theory of the coupling of two angular momentum states [32, 33]. 

The final results contain two terms which are respectively angular dependent and angular independent. We shall discuss the latter first, as it corresponds to the situation where there is no preferred orientation in the laboratory frame for the appropriate principal axis, i.e. the absorber is a non-oriented polycrystalline material. 

If the y-ray transition is between two levels of nuclear spin and /j, and furthermore between the two substates with 7 values of nii and ntz respectively, then the angular-independent probabihty term is given by the square of the appropriate Clebsch-Gordan coefficients [32]. 

J is the vector sum J = Ii + Iz, and m is the vector sum m = nti — ntz. /is also known as the multipolarity of the transition, and the smaller values of J give the larger intensities. / = 1 is a dipole transition and / = 2 is a quadrupole transition, etc. If there is no change in parity during the decay it is classified as magnetic dipole (Ml) or electric quadrupole (E2). Electric dipole (El) transitions with a change in parity also come within our scope. In some cases the y decay is a mixed dipole-quadrupole radiation, so that both must be included in the calculations. 

Most Mossbauer isotopes decay primarily by a dipole transition. This puts an effective bar on the number of values which m can adopt. All values of m other than 0, 1 cause the coefficient to be zero. For quadrupole transitions, m can be 0, 1, 2. The majority of the Mossbauer isotopes have ground/excited state spins which are one of the following 0,2  

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To calculate the relative intensities of absorption lines in a Mossbauer spectrum, one also has to take into account that the intensity of the emitted multipole radiation is not isotropic but has a certain angular dependence Fim 0) where 6 is the polar angle that the direction of observation encloses with the z-axis of the principal axis system (defined, e.g., by the direction... 

The effect of an anisotropic recoil-free fraction on the relative intensities of absorption lines in hyperflne split spectra has been observed and interpreted in Fe (Goldanskii, Makarov Khrapov, 1963) and the effect has subsequently been observed in almost all Mossbauer isotopes. Particularly large effects have been seen with Eu in Eu2Ti207 (Armon et ai, 1973 Bauminger et al., 1974) and examples are shown in Figure 6.5. 

Figure 3.3 shows some of these possible transitions for HCI. Those with A7 = +1 are known as the R branch and occur at the high-energy side of the hypothetical transition At = 1, A7 = 0 (this is not allowed because of the selection rule, A7 = +1). Those with A7 = — 1 on the low-frequency side of the hypothetical transition form the P branch. Figure 3.4 shows the absorption spectrum of HCI at room temperature, with the rotational transitions responsible for each line. The relative intensities of the lines reflect the relative populations of the absorbing rotational levels the peaks are doublets due to the separate absorptions of the two chlorine isotopes, that is, H35C1 and H37C1, which have different reduced masses and hence values of the rotational constant B. 

Relative intensities of spectral lines, if they are accurately known, also can be used for photographic emulsion calibration. For this purpose, it is necessary that the relative spectral line intensities be well established and that they exhibit no self-absorption. The lines should have approximately the same excitation energies. The spectral lines should lie in the same general spectral region since emulsion contrast (emulsion gamma) varies with wavelength. 

Although the excitation spectrum directly reflects the absorption spectrum with respect to the line positions, the relative intensities of different lines I X) are identical in both spectra only if the following conditions are guaranteed ... 

The relative intensities of rotational lines in a vibrational band of an electronic absorption spectrum are almost entirely dependent on the relative populations on the initial rotational levels (Maxwell-Boltzman distribution). Use this fact to make a sketch of fhe Q-branches in Example 10.1 assuming a sample temperature of 300K. How would fhey appear under low resolution ... 

The method involves the irradiation of a sample with polychromatic X-rays (synchrotron radiation) which inter alia promote electrons from the innermost Is level of the sulfur atom to the lowest unoccupied molecular orbitals. In the present case these are the S-S antibonding ct -MOs. The intensity of the absorption lines resulting from these electronic excitations are proportional to the number of such bonds in the molecule. Therefore, the spectra of sulfur compounds show significant differences in the positions and/or the relative intensities of the absorption lines [215, 220, 221]. In principle, solid, liquid and gaseous samples can be measured. 

The XANES spectra of Se, Ss, Sio, S12, S14 [222] and polymeric sulfur [223, 224] are all very similar as far as the peak positions are concerned but the relative intensities of the two peaks differ considerably. The spectra are characterized by an absorption line at 2471.7 eV (so-called white line) and a broad absorption in the region 2477-2480 eV [222] see Fig. 35 (energies calibrated to the white line of ZnS04 defined as 2481.4 eV). Since the spectra of the components of a mixture are additive, quantitative analyses are possible, even for mixtures of samples as similar as Ss and polymeric sulfur, for instance [224]. The interpretation of the spectra is, however, still somewhat controversial see [225]. 

Thus, we can predict the relative intensity of each 0 m absorption line using this expression together with Equation (5.30) ... 

Excitation wavelength dependence of the Raman lines presents another complication that is not a problem for transient absorption measurements. For example, in comparing spectra taken at two different excitation wavelengths, one must consider not only which and in what proportion each molecular species is pumped, but also keep in mind what differences in relative intensity of the Raman lines result from excitation at the two exciting wavelengths. The complication can usually be overcome, especially when the spectra of the two species are well resolved. 

Analysis of the microwave spectrum of piperidine and of A-deuteropiper-idine suggests that the strongest Q-branch series together with the associated R-branch lines arise from the N-H-axial conformer.144 From this absorption and from a weaker series of Q branches, /eq//aj (the relative intensities of the type-A lines of /V-Hcq and N-Hax conformers corrected for the frequency factor) was estimated as 1/6 at — 34°C. This ratio is related to AE = En Hax - jv-Hcq by the expression... 

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