Nuclear states lifetimes

If the spin state interconversion is faster than the excited nuclear state lifetime, that is x 10 7 second, then the observed spectrum is an average of the spectra of the two spin states. Until recently this condition had been observed only for iron(III) complexes with thiocar-bamate or selenocarbamate ligands—ferric dithiocarbamates (119), monothiocarbamates (98), or diselenocarbamates (42). Since 1982, however, there have been a number of reports of other iron(III) complexes which also display an averaged Mossbauer spectrum (56, 57, 108 111, 124, 153, 155). 

In the early days of TCSPC the moment or centroid shift method was used to calculate fluorescence and excited nuclear state lifetimes from measurement data [348, 549]. For fluorescence lifetime measurements this method has now entirely been replaced with curve fitting procedures. However, the first-moment technique has recently been revived for use in functional brain imaging [325]. A benefit of the first-moment technique is that the statistical accuracy of r calculated via the first moments can be easily estimated. Let the standard deviation of the photon arrival times in the IRF recording be standard deviation of the first... 

Projecting the nuclear solutions Xt( ) oti the Hilbert space of the electronic states (r, R) and working in the projected Hilbert space of the nuclear coordinates R. The equation of motion (the nuclear Schrddinger equation) is shown in Eq. (91) and the Lagrangean in Eq. (96). In either expression, the terms with represent couplings between the nuclear wave functions X (K) and X (R). that is, (virtual) transitions (or admixtures) between the nuclear states. (These may represent transitions also for the electronic states, which would get expressed in finite electionic lifetimes.) The expression for the transition matrix is not elementaiy, since the coupling terms are of a derivative type. 

The description of states participating in a spin-state transition as electronic isomers with discrete nuclear configurations, in particular different metal-ligand distances, requires that separate electronic and vibrational spectra of the two spin states exist. Indeed, a superposition of the individual vibrational spectra of the two states is in general observed, the relative contribution of the states being a function of temperature [41, 139, 140, 141, 142]. This observation sets a lower limit for the spin-state lifetime longer than the nuclear vibrational period, i.e.,... 

On the other hand, NMR spectra appear in general as the average of the spectra of the two spin states [36, 153]. This observation determines an upper limit for the spin-state lifetime shorter than the nuclear spin relaxation time Tl = l/ktH < lO s. In general, therefore, either the superposition or the average of the particular spectroscopic properties of the two spin states is observed, subject to the relative magnitude of lifetime of the excited spectroscopic state and the rate of spin-state conversion. The rate /clh is thus estimated... 

A unique situation is encountered if Fe-M6ssbauer spectroscopy is applied for the study of spin-state transitions in iron complexes. The half-life of the excited state of the Fe nucleus involved in the Mossbauer experiment is tj/2 = 0.977 X 10 s which is related to the decay constant k by tj/2 = ln2/fe. The lifetime t = l//c is therefore = 1.410 x 10 s which value is just at the centre of the range estimated for the spin-state lifetime Tl = I/Zclh- Thus both the situations discussed above are expected to appear under suitable conditions in the Mossbauer spectra. The quantity of importance is here the nuclear Larmor precession frequency co . If the spin-state lifetime Tl = 1/feLH is long relative to the nuclear precession time l/co , i.e. Tl > l/o) , individual and sharp resonance lines for the two spin states are observed. On the other hand, if the spin-state lifetime is short and thus < l/o) , averaged spectra with intermediate values of quadrupole splitting A q and isomer shift 5 are found. For the intermediate case where Tl 1/cl , broadened and asymmetric resonance lines are obtained. These may be the subject of a lineshape analysis that will eventually produce values of rate constants for the dynamic spin-state inter-conversion process. The rate constants extracted from the spectra will be necessarily of the order of 10 -10 s"F... 

The ratio F/Eq of width F and the mean energy of the transition Eo defines the precision necessary in nuclear y-absorption for tuning emission and absorption into resonance. Lifetimes of excited nuclear states suitable for Mossbauer spectroscopy range from 10 s to s. Lifetimes longer than 10 s produce too... 

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. 

The nuclear y-resonance effect in ° Ni was first observed in 1960 by Obenshain and Wegener [2]. However, the practical application to the study of nickel compounds was hampered for several years by (1) the lack of a suitable single-line source, (2) the poor resolution of the overlapping broad hyperfine lines due to the short excited state lifetime, and (3) the difficulties in producing and handling the short-lived Mossbauer sources containing the Co and Cu parent nuclides, respectively. 

The nuclear decay of radioactive atoms embedded in a host is known to lead to various chemical and physical after effects such as redox processes, bond rupture, and the formation of metastable states [46], A very successful way of investigating such after effects in solid material exploits the Mossbauer effect and has been termed Mossbauer Emission Spectroscopy (MES) or Mossbauer source experiments [47, 48]. For instance, the electron capture (EC) decay of Co to Fe, denoted Co(EC) Fe, in cobalt- or iron-containing compormds has been widely explored. In such MES experiments, the compormd tmder study is usually labeled with Co and then used as the Mossbauer source versus a single-line absorber material such as K4[Fe(CN)6]. The recorded spectrum yields information on the chemical state of the nucleogenic Fe at ca. 10 s, which is approximately the lifetime of the 14.4 keV metastable nuclear state of Fe after nuclear decay. 

As was already noted in [9], the primary effect of the YM field is to induce transitions (Cm —> Q) between the nuclear states (and, perhaps, to cause finite lifetimes). As already remarked, it is not easy to calculate the probabilities of transitions due to the derivative coupling between the zero-order nuclear states (if for no other reason, then because these are not all mutually orthogonal). Efforts made in this direction are successful only under special circumstances, for example, the perturbed stationary state method [64,65] for slow atomic collisions. This difficulty is avoided when one follows Yang and Mills to derive a mediating tensorial force that provide an alternative form of the interaction between the zero-order states and, also, if one introduces the ADT matrix to eliminate the derivative couplings. 

All nuclear multiplet structures due to coupling of nonequivalent nuclei are, as noted earlier, subject to effects on line shapes by chemical or positional exchange. For those multiplet structures arising from coupling of nuclei, one of which has a nonzero nuclear quadrupole moment, effects of quadrupole relaxation must be considered. For example, if a proton or fluorine atom is bonded to a nitrogen nucleus (I = 1), a triplet resonance will be expected in the proton or fluorine spectrum. For observation of this fine structure it is necessary that the lifetimes of the nuclear spin states of nitrogen (m = 1, 0, —1) be greater than the inverse frequency separation between multiplet components, i.e., t > l/ANx (106). The lifetimes of N14 spin states can become comparable to or less than 1 /A as a result of quadrupole relaxation. When the N14 spin-state lifetimes are comparable... 

One should remember in this connection that many cases of spontaneous fission of excited (isomer) nuclear states with lifetimes about 10 2 sec (quite close to the time of addition of a new link to the growing polyformaldehyde chain near absolute zero of temperature) were observed and successfully studied after the pioneering works of Dubna scientists. 

The two techniques which have been used effectively to set limits on the rates of spin state interconversions are Mossbauer and EPR spectroscopies. As described in Section III,E, the lifetime of the excited nuclear state involved in the Mossbauer effect is 10"7 second. Thus the observation of the Mossbauer spectrum can immediately classify the spin state lifetime as greater than or less than 10"7 second. Both conditions have been observed, as was described in Section III,E. 

This methodology has been applied in many areas, such as the measurement of lifetimes of excited nuclear states and nuclear magnetic moments, the investigation of electric and magnetic fields in atoms and crystals, in the analysis of special relativity, the equivalence principle, and also in other applications [50-57],... 

Figure 1 depicts a typical sequence of events started by absorption of an incident photon with an energy near the nuclear excited state energy Eq. The Fe nucleus has an excited state lifetime of 141ns, and excited nuclei have two decay channels. About 10% of them reemit a 14.4kev photon. For recoilless absorption, where no vibrational levels are excited, time-resolved measurements of 14.4kev photons scattered in the forward direction reveal information on hyperfine interactions comparable to conventional Mossbauer spectroscopy (see Mossbauer Spectroscopy). The remaining nuclei expel electrons from the atomic K shell, followed by... 

PPE signals give direct information on the density of states of the unoccupied states which is obtained only indirectly with other optical methods. One drawback is that since the excited electrons are detected, the observation time window is limited to the lifetime of the excited electrons. The excited state lifetimes at metal surfaces are typically less than a few hundreds of femtoseconds and much shorter than vibrational relaxation times. Hence the information is limited to that in the very beginning of the nuclear wavepacket motion, right after the photoexcitation. 

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