Quantum number nuclear spin

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

THE SCHRODINGER EQUATION AND SOME OF ITS SOLUTIONS Table 1.3 Some values of the nuclear spin quantum number / 19... 

Similar principles apply to ortho- and para-deuterium except that, as the nuclear spin quantum number of the deuteron is 1 rather than as for the proton, the system is described by Bose-Einstein statistics rather than the more familiar Eermi-Dirac statistics. Eor this reason, the stable low-temperature form is orriio-deuterium and at high temperatures the statistical weights are 6 ortho 3 para leading to an upper equilibrium concentration of 33.3% para-deuterium above about 190K as shown in Eig. 3.1. Tritium (spin 5) resembles H2 rather than D2. 

Phosphorus has only one stable isotope, J P, and accordingly (p. 17) its atomic weight is known with extreme accuracy, 30.973 762(4). Sixteen radioactive isotopes are known, of which P is by far the most important il is made on the multikilogram scale by the neutron irradiation of S(n,p) or P(n,y) in a nuclear reactor, and is a pure -emitter of half life 14.26 days, 1.7()9MeV, rntan 0.69MeV. It finds extensive use in tracer and mechanistic studies. The stable isotope has a nuclear spin quantum number of and this is much used in nmr spectroscopy. Chemical shifts and coupling constants can both be used diagnostically to determine structural information. 

The isotope has a nuclear spin quantum number I and so is potentially useful in nmr experiments (receptivity to nmr detection 17 X 10 that of the proton). The resonance was first observed in 1951 but the low natural abundance i>i S(0.75%) and the quadrupolar broadening of many of the signals has so far restricted the amount of chemically significant work appearing on this rcsonance, However, more results are expected now that pulsed fourier-transform techniques have become generally available. 

Receptivity D is proportional to y NKJ + 1) where y is the magnetogyric ratio, N the natural abundance of the isotope, and / the nuclear spin quantum number Dp is the receptivity relative to that of the proton taken as 1.000. 

All have zero nuclear spin except (33.8% abundance) which has a nuclear spin quantum number this isotope finds much use in nmr spectroscopy both via direct observation of the Pt resonance and even more by the observation of Pt satellites . Thus, a given nucleus coupled to Pt will be split into a doublet symmetrically placed about the central unsplit resonance arising from those species containing any of the other 5 isotopes of Pt. The relative intensity of the three resonances will be (i X 33.8) 66.2 ( x 33.8), i.e. 1 4 1. 

We often say that an electron is a spin-1/2 particle. Many nuclei also have a corresponding internal angular momentum which we refer to as nuclear spin, and we use the symbol I to represent the vector. The nuclear spin quantum number I is not restricted to the value of 1/2 it can have both integral and halfintegral values depending on the particular isotope of a particular element. All nuclei for which 7 1 also posses a nuclear quadrupole moment. It is usually given the symbol Qn and it is related to the nuclear charge density Pn(t) in much the same way as the electric quadrupole discussed earlier ... 

The nuclear spin product functions will be represented by where n represents a given nuclear spin configuration characterized by nuclear spin quantum number Wj and for nuclei j and k, respectively. 

Nuclear spin quantum number of ground (g) and excited (e) state (the sign refers to the parity)... 

This equation is called the Curie law and relates the equilibrium magnetization M0 to the strength of the magnetic field B0. The constants have the following meaning I is the nuclear spin quantum number (see below), y is the gyromagnetic ratio specific for a given isotope, h is Planck s constant, kB is Boltzmann s constant, N is the number of nuclei and T is the temperature. 

Hz Hertz, unit of frequency (cycles per m (1) Nuclear spin quantum number ... 

The eigenfunctions of the spin Hamiltonian [eqn (1.7)] are expressed in terms of an electron- and nuclear-spin basis set ms, mr), corresponding to the electron and nuclear spin quantum numbers ms and mr, respectively. The energy eigenvalues of eqn (1.7) are ... 

Equation (2.3) describes line positions correctly for spectra with small hyperfine coupling to two or more nuclei provided that the nuclei are not magnetically equivalent. When two or more nuclei are completely equivalent, i.e., both instantaneously equivalent and equivalent over a time average, then the nuclear spins should be described in terms of the total nuclear spin quantum numbers I and mT rather than the individual /, and mn. In this coupled representation , the degeneracies of some multiplet lines are lifted when second-order shifts are included. This can lead to extra lines and/or asymmetric line shapes. The effect was first observed in the spectrum of the methyl radical, CH3, produced by... 

Given the Hamiltonian eqn (3.1), it is reasonable to express the eigenfunctions in terms of the electron and nuclear spin quantum numbers ms,mi). Applying to this function only the two terms in the Hamiltonian operator that involve the -direction of the field B we get ... 

Methyl radicals formed on a silica gel surface are apparently less mobile and less stable than on porous glass (56, 57). The spectral intensity is noticeably reduced if the samples are heated to —130° for 5 min. The line shape is not symmetric, and the linewidth is a function of the nuclear spin quantum number. Hence, the amplitude of the derivative spectrum does not follow the binomial distribution 1 3 3 1 which would be expected for a rapidly tumbling molecule. A quantitative comparison of the spectrum with that predicted by relaxation theory has indicated a tumbling frequency of 2 X 107 and 1.3 X 107 sec-1 for CHr and CD3-, respectively (57). 

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