Nuclear state

As was already noted in [9], the primary effect of the YM field is to induce transitions between the nuclear states (and, perhaps, to cause finite... 

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. 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

In Chapter IV, Englman and Yahalom summarize studies of the last 15 years related to the Yang-Mills (YM) field that represents the interaction between a set of nuclear states in a molecular system as have been discussed in a series of articles and reviews by theoretical chemists and particle physicists. They then take as their starting point the theorem that when the electronic set is complete so that the Yang-Mills field intensity tensor vanishes and the field is a pure gauge, and extend it to obtain some new results. These studies throw light on the nature of the Yang-Mills fields in the molecular and other contexts, and on the interplay between diabatic and adiabatic representations. 

The results of the theory of quantum mechanics require that nuclear states have discrete energies. This is in contrast to classical mechanical systems, which can have any of a continuous range of energies. This difference is a critical fact in the appHcations of radioactivity measurements, where the specific energies of radiations are generally used to identify the origin of the radiation. Quantum mechanics also shows that other quantities have only specific discrete values, and the whole understanding of atomic and nuclear systems depends on these discrete quantities. 

These rules have very distinctinfluences on the decays of nuclear states. 

Half-Lives and Decay Constants. Each nuclear state, whether an unstable ground state or an excited level, has a characteristic probabiUty... 

AIterna.tives to y-Ray Emission. y-Ray emission results ia the deexcitation of an excited nuclear state to a lower state ia the same nucHde, ie, no change ia Z or. There are two other processes by which this transition can take place without the emission of a y-ray of this energy. These are internal conversion and internal pair formation. The internal-conversion process iavolves the transfer of the energy to an atomic electron. 

Internal Conversion. As an alternative to the emission of a y-ray, the available energy of the excited nuclear state can be transferred to an atomic electron and this electron can then be ejected from the atom. The kinetic energy of this electron is where E is the energy by which the... 

The emission of y rays follows, in the majority of cases, what is known as P decay. In the P-decay process, a radionuclide undergoes transmutation and ejects an electron from inside the nucleus (i.e., not an orbital electron). For the purpose of simplicity, positron and electron capture modes are neglected. The resulting transmutated nucleus ends up in an excited nuclear state, which prompdy relaxes by giving offy rays. This is illustrated in Figure 2. 

Now let a steady field be applied. The two nuclear states now correspond to orientation of the bar magnet parallel to the field (i.e., N pole to S pole) or antiparallel to the field (N pole to N pole). There will be an energy difference between these states, the orientation with the field (N to S) being of lower energy than the orientation against the field. 

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. 

So far, we have discussed only the detection of y-rays transmitted through the Mossbauer absorber. However, the Mossbauer effect can also be established by recording scattered radiation that is emitted by the absorber nuclei upon de-excitation after resonant y-absorption. The decay of the excited nuclear state proceeds for Fe predominantly by internal conversion and emission of a conversion electron from the K-shell ( 90%). This event is followed by the emission of an additional (mostly Ka) X-ray or an Auger electron when the vacancy in the K shell is filled again. Alternatively, the direct transition of the resonantly excited nucleus causes re-emission of a y-photon (14.4 keV). 

Quadmpole interaction lifts the degeneracy of nuclear states with spin quantum numbers I > 1/2, and is manifested in the Mossbauer spectmm as quadmpole splitting A q (as will be further discussed in Sect. 4.3). According to (4.5), the classical electric monopole and quadmpole interaction energies Ei and q are additive, that is, = E + Eq. 

The electric monopole interaction between a nucleus (with mean square radius k) and its environment is a product of the nuclear charge distribution ZeR and the electronic charge density e il/ 0) at the nucleus, SE = const (4.11). However, nuclei of the same mass and charge but different nuclear states isomers) have different charge distributions ZeR eR ), because the nuclear volume and the mean square radius depend on the state of nuclear excitation R R ). Therefore, the energies of a Mossbauer nucleus in the ground state (g) and in the excited state (e) are shifted by different amounts (5 )e and (5 )g relative to those of a bare nucleus. It was recognized very early that this effect, which is schematically shown in Fig. 4.1, is responsible for the occurrence of the Mossbauer isomer shift [7]. 

Fig. 4.1 The electric monopole interaction between the nuclear charge and the electron density at the nucleus shifts the energy of the nuclear states and gives the Mossbauer isomer shift...

Fig. 4.6 Quadrupole splitting of the excited state of Fe with I = 3/2 and the resulting Mossbauer spectrum. Quadrupole interaction splits the spin quartet into two degenerate sublevels 7, OT/) with energy separation A q = 2 q. The ground state with I = 1/2 remains unsplit. The nuclear states are additionally shifted by electric monopole interaction giving rise to the isomer shift 8...

Fig. 4.9 Magnetic dipole splitting (nuclear Zeeman effect) in pe and resultant Mossbauer spectrum (schematic). The mean energy of the nuclear states is shifted by the electric monopole interaction which gives rise to the isomer shift 5. Afi. g = Sg/tN and A M,e = refer to the...

Pure nuclear magnetic hyperfine interaction without electric quadrupole interaction is rarely encountered in chemical applications of the Mossbauer effect. Metallic iron is an exception. Quite frequently, a nuclear state is perturbed simultaneously by... 

If the electric quadrupole splitting of the 7 = 3/2 nuclear state of Fe is larger than the magnetic perturbation, as shown in Fig. 4.13, the nij = l/2) and 3/2) states can be treated as independent doublets and their Zeeman splitting can be described independently by effective nuclear g factors and two effective spins 7 = 1/2, one for each doublet [67]. The approach corresponds exactly to the spin-Hamiltonian concept for electronic spins (see Sect. 4.7.1). The nuclear spin Hamiltonian for each of the two Kramers doublets of the Fe nucleus is ... 

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