Spin nuclear

A nucleus with non-zero spin acts as a magnetic dipole, giving raise to a vector potential Aa. 

Here Iais the magnetic moment of nucleus A and Ra is the position (the nucleus is the natural Gauge origin). Adding this to the external vector potential in eq. (10.62) and expanding as in (10.63) gives 

All the operators P2, P and L are gauge dependent, relating to the position of atom A via eqs. (10.9) and (10.74), and each of the dia- and para-magnetic terms depends on the chosen gauge. 

In order to describe nuclear spin-spin coupling, we need to include electron and nuclear spins, which are not present in the non-relativistic Hamilton operator. A relativistic treatment, as shown in Section 8.2, gives a direct nuclear-nuclear coupling term (eq. (8.33)). 

For rapidly tumbling molecules (solution or gas phase) this contribution averages out to zero, but it is significant for solid state NMR. The operator (eq. (8.32)) gives an indirect term, which is normally written as two separate operators. 

The and operators determine the isotropic and anisotropic parts of the hyperfine coupling constant (eq. (10.11)), respectively. The latter contribution averages out for rapidly tumbling molecules (solution or gas phase), and the (isotropic) hyperfine coupling constant is therefore determined by the Fermi-Contact contribution, i.e. the electron density at the nucleus. 

The indirect spin-spin coupling between nuclei A and B, which is the one observed in solution phase NMR, contains several contributions, as seen from eq. (10.18)----------------- 

When placed in a static magnetic field of strength Hq, the magnetic moment, fx, of the spinning nucleus precesses about the magnetic field at a 

The amount of separation of the two energy states, AE, is proportional to the magnetic field, and is given by the following expression  

When nuclei in the magnetic field are exposed to radiation of the proper frequency, transitions between the two energy states are stimulated, and the nucleus is said to be in resonance, or to resonate. This transition occurs when the frequency and the energy difference are related by the Planck relation, E = hv, and thus the sample will absorb energy of frequency The study of these energy changes is known as nuclear magnetic resonance, or NMR, spectroscopy. 

Of course, it would be very difficult to determine the frequencies of many superimposed cosine waves from this kind of plot, and it would be at best awkward to interpret a complex NMR spectmm presented in such a fashion (Fig. 8.10). The use of the Fourier transform allows us mathematically to 

The magnitude of the chemical shift depends on the nature of the valence and inner electrons of the nucleus and even on electrons that are not directly associated with the nucleus. Chemical shifts are influenced by inductive effects, which reduce the electron density near the nucleus and reduce the shielding. The orientation of the nucleus relative to tt electrons also plays an important role in determining the chemical shift. A proton located immediately outside a TT-electron system (as in the case of the protons on benzene rings) will be significantly deshielded. In most molecules the chemical shift is determined by a combination of these factors. Chemical shifts are difficult to predict using theoretical principles, but have been weU studied and can usually be easily predicted empirically upon comparison to reference data. 

Hydrogen comes in three isotopes protium (]H or H), deuterium ( H or D), and tritium ( H or T). They all contain one proton, which, by definition, makes them hydrogen then they possess either 0,1, or 2 neutrons, respectively. 

Just as the cation of protium is called a proton, the cation of deuterium is called a deuteron, and the cation of tritium is called a triton. Collectively, a hydrogen ion of an unknown isotope is simply referred to as a hydron. 

Protium is by far the most abundant at 99.986 percent. Deuterium is known as heavy hydrogen and has an abundance of 0.014 percent tritium is a radioactive material with an abundance of 7x10 percent on earth however, it s been proposed that larger amounts of tritium exist on the moon. Both deuterium and tritium undergo very exothermic fusion reactions, so they are used in both research and thermonuclear weapons. There s also the hopeful possibility that they might be used as a future energy source also. 

All the isotopes have the same electronic configuration, so they exhibit identical chemistry however, the reactions occur at different rates. For example, the adsorption of H2 to surfaces is more rapid than that of D2. And H2 reacts 13 times faster with CI2 than D2, because H2 has a lower activation energy. 

The geATRS - B term in eq. (10.68) in connection with Ba in eq. (10.66) gives three terms, which conventionally are collected in two operators. 

H is a (one electron) Spin-Dipolar and H e is a Fermi Contact operator, and their sum is the operator in eq. (8.37). 

The Aa P term in eq. (10.68) gives the Paramagnetic Spin-Orbit operator. 

The V2AA term in eq. (10.68) gives a Diamagnetic Spin-Orbit operator, which is an operator of order c. Although we otherwise only consider terms up to order c, the nuclear spin-spin coupling constant only contains terms of order c, which is why we need to include 

When both nuclear spins and an external magnetic field are present, there is an additional mixed Aext Aa term arising from the expansion of the generalized momentum operator. 

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In nuclear spectroscopy fine structure arises from coupling between nuclear spins. 

In the following pages, only hydrogen and the carbon 13 isotope, both of which have a nuclear spin of 1/2 will be discussed. 

NMR Nuclear magnetic resonance [223, 224] Chemical shift of splitting of nuclear spin states in a magnetic field H [225], C [226, 227], N [228], F [229], 2 Xe [230] Other Techniques Chemical state diffusion of adsorbed species... 

Initially, we neglect tenns depending on the electron spin and the nuclear spin / in the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be N(see (equation Al.4.1)) which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of. m the (X, Y, Z) axis system are given by ... 

Finally, we consider the complete molecular Hamiltonian which contains not only temis depending on the electron spin, but also temis depending on the nuclear spin / (see chapter 7 of [1]). This Hamiltonian conmiutes with the components of Pgiven in (equation Al.4,1). The diagonalization of the matrix representation of the complete molecular Hamiltonian proceeds as described in section Al.4,1.1. The theory of rotational synnnetry is an extensive subject and we have only scratched the surface here. A relatively new book, which is concemed with molecules, is by Zare [6] (see [7] for the solutions to all the problems in [6] and a list of the errors). This book describes, for example, the method for obtaining the fimctioiis ... 

These limitations lead to electron spin multiplicity restrictions and to differing nuclear spin statistical weights for the rotational levels. Writing the electronic wavefunction as the product of an orbital fiinction and a spin fiinction there are restrictions on how these functions can be combined. The restrictions are imposed by the fact that the complete function has to be of synnnetry... 

The way we combine the nuclear spin basis fiinctions with the rotation-vibration-electronic basis fiinctions in H2 follows the same type of argument iism the nuclear penniitation group Rovibronic states of synnnetry... 

Labelling the O nuclei 1 and 2 this group is as in table Al.4.7. The spin of nuclei is 0 and so the nuclear spin wavefiinction is of species There is no... 

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. 

Cordonnier M, Uy D, Dickson R M, Kew K E, Zhang Y and Oka T 2000 Selection rules for nuclear spin modifications in ion-neutral reactions involving Hg" J. Chem. Phys. 113 3181-93... 

NMR depends on manipulating the collective motions of nuclear spins, held in a magnetic field. As with... 

Therefore, in NMR, one observes collective nuclear spin motions at the Lannor frequency. Thus the frequency of NMR detection is proportional to Nuclear magnetic moments are connnonly measured either... 

Harris R K 1996 Nuclear spin properties and notation Encyclopedia of NMRvo 5, ed D M Grant and R K Harris (Chichester Wiley) pp 3301-14... 

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 ... 

Table Bl.13.2 Interactions giving rise to nuclear spin relaxation.

Woessner D E 1962 Nuclear spin relaxation in ellipsoids undergoing rotational Brownian motion J. Chem. Rhys. 37 647-54... 

Szymanski S, Gryff-Keller A M and Binsch G A 1986 Liouville space formulation of Wangsness-Bloch-Redfield theory of nuclear spin irelaxation suitable for machine computation. I. Fundamental aspects J. Magn. Reson. 68 399-432... 

Maler L, Widmalm G and Kowalewski J 1996 Dynamical behavior of carbohydrates as studied by carbon-13 and proton nuclear spin relaxation J. Phys. Chem. 100 17 103-10... 

Dayie K T, Wagner G and Lefeevre J F 1996 Theory and practice of nuclear spin relaxation in proteins Anna. Rev. Phys. Chem. 47 243-82... 

Kowalewski J 1990 Nuclear spin relaxation in diamagnetic fluids Part 1 Annu. Rep. NMR Spectrosc. 22 308-414 1991 Part 2 Annu. Rep. NMR Spectrosc. 23 289-374... 

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... 

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