Energy level, nuclear

IV. The Measurement of Nuclear Energy Levels—Nuclear Quadrupole Resonance... 

IV. THE MEASUREMENT OF NUCLEAR ENERGY LEVELS— NUCLEAR QUADRUPOLE RESONANCE AND THE MOSSBAUER EFFECT... 

Atomic size, highest occupied principal energy level, nuclear charge... 

Nuclear magnetic resctnance involves the transitions between energy levels of the fourth quantum number, the spin quantum number, and only certain nuclei whose spin is not zero can be studied by this technique. Atoms having both an even number of protons and neutrons have a zero spin for example, carbon 12, oxygen 16 and silicon 28. 

It would appear that identical particle pemuitation groups are not of help in providing distinguishing syimnetry labels on molecular energy levels as are the other groups we have considered. However, they do provide very usefiil restrictions on the way we can build up the complete molecular wavefiinction from basis fiinctions. Molecular wavefiinctions are usually built up from basis fiinctions that are products of electronic and nuclear parts. Each of these parts is fiirther built up from products of separate uncoupled coordinate (or orbital) and spin basis fiinctions. Wlien we combine these separate fiinctions, the final overall product states must confonn to the pemuitation syimnetry mles that we stated above. This leads to restrictions in the way that we can combine the uncoupled basis fiinctions. 

The rotation-vibration-electronic energy levels of the PH3 molecule (neglecting nuclear spin) can be labelled with the irreducible representation labels of the group The character table of this group is given in table Al.4.10. 

Of the NMR-active nuclei around tluee-quarters have / > 1 so that the quadnipole interaction can affect their spectra. The quadnipole inter action can be significant relative to the Zeeman splitting. The splitting of the energy levels by the quadnipole interaction alone gives rise to pure nuclear quadnipole resonance (NQR) spectroscopy. This chapter will only deal with the case when the quadnipole interaction can be regarded as simply a perturbation of the Zeeman levels. 

Nuclear spin relaxation is caused by fluctuating interactions involving nuclear spins. We write the corresponding Hamiltonians (which act as perturbations to the static or time-averaged Hamiltonian, detemiming the energy level structure) in tenns of a scalar contraction of spherical tensors ... 

The interaction of the electron spin s magnetic dipole moment with the magnetic dipole moments of nearby nuclear spins provides another contribution to the state energies and the number of energy levels, between which transitions may occur. This gives rise to the hyperfme structure in the EPR spectrum. The so-called hyperfme interaction (HFI) is described by the Hamiltonian... 

Figure Bl.15.8. (A) Left side energy levels for an electron spin coupled to one nuclear spin in a magnetic field, S= I =, gj >0, a<0, and a l 2h)<(a. Right side schematic representation of the four energy levels with )= Mg= , Mj= ). +-)=1, ++)=2, -)=3 and -+)=4. The possible relaxation paths are characterized by the respective relaxation rates W. The energy levels are separated horizontally to distinguish between the two electron spin transitions. Bottom ENDOR spectra shown when a /(21j)< ca (B) and when co < a /(2fj) (C).

The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. 

It turns out though that there are several possible variations on this general theme We could for example keep the magnetic field constant and continuously vary the radiofrequency until it matched the energy difference between the nuclear spin states Or we could keep the rf constant and adjust the energy levels by varying the magnetic... 

Decay Schemes. Eor nuclear cases it is more useful to show energy levels that represent the state of the whole nucleus, rather than energy levels for individual atomic electrons (see Eig. 2). This different approach is necessary because in the atomic case the forces are known precisely, so that the computed wave functions are quite accurate for each particle. Eor the nucleus, the forces are much more complex and it is not reasonable to expect to be able to calculate the wave functions accurately for each particle. Thus, the nuclear decay schemes show the experimental levels rather than calculated ones. This is illustrated in Eigure 4, which gives the decay scheme for Co. Here the lowest level represents the ground state of the whole nucleus and each level above that represents a different excited state of the nucleus. 

The technique for measurement which is most easily interpreted is the inversion-recovery method, in which the distribution of the nuclear spins among the energy levels is inverted by means of a suitable 180° radiofrequency pulse A negative signal is observed at first, which becomes increasingly positive with time (and hence also with increasing spin-lattice relaxation) and which... 

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